{"id":34,"date":"2023-09-17T11:01:40","date_gmt":"2023-09-17T11:01:40","guid":{"rendered":"https:\/\/mathority.org\/pt\/derivada-de-uma-subtracao\/"},"modified":"2023-09-17T11:01:40","modified_gmt":"2023-09-17T11:01:40","slug":"derivada-de-uma-subtracao","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/derivada-de-uma-subtracao\/","title":{"rendered":"Derivada de uma subtra\u00e7\u00e3o"},"content":{"rendered":"<p>Neste artigo explicamos como derivar uma subtra\u00e7\u00e3o de fun\u00e7\u00f5es (f\u00f3rmula). Voc\u00ea tamb\u00e9m encontrar\u00e1 exemplos de derivadas de subtra\u00e7\u00e3o e exerc\u00edcios resolvidos passo a passo para praticar. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-derivada-de-una-resta\"><\/span> F\u00f3rmula para a derivada de uma subtra\u00e7\u00e3o<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>A derivada da subtra\u00e7\u00e3o de duas fun\u00e7\u00f5es \u00e9 igual a subtrair a derivada de cada fun\u00e7\u00e3o separadamente.<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2eb280531ada831cf451f6440ac3cb7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z(x)=f(x)-g(x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} z'(x)=f'(x)-g'(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"463\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Por outras palavras, diferenciar duas fun\u00e7\u00f5es separadamente e depois subtra\u00ed-las equivale a primeiro subtrair as fun\u00e7\u00f5es e depois calcular a derivada. <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/derivee-d-une-soustraction-1.webp\" alt=\"derivada de uma subtra\u00e7\u00e3o\" class=\"wp-image-2053\" width=\"243\" height=\"243\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Da mesma forma, a mesma regra de diferencia\u00e7\u00e3o se aplica a subtra\u00e7\u00f5es de duas ou mais fun\u00e7\u00f5es, portanto, se tivermos uma subtra\u00e7\u00e3o de tr\u00eas, quatro, cinco,\u2026 fun\u00e7\u00f5es, precisamos derivar cada uma separadamente e depois subtra\u00ed-las.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8bc282473a7f7560eebc97db66bb8f54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}z(x)=f(x)- g(x)- h(x)- \\dots\\\\[1.5ex]\\color{orange}\\bm{\\downarrow}\\quad\\color{black} \\\\[1.5ex]z'(x)=f'(x)-g'(x)- h'(x)- \\dots\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"261\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Como voc\u00ea pode ver, a f\u00f3rmula da derivada da diferen\u00e7a de fun\u00e7\u00f5es \u00e9 muito semelhante \u00e0 regra da derivada da soma.<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Veja:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/derivada-de-uma-soma\/\">derivada de uma soma de fun\u00e7\u00f5es<\/a><\/span> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplos-de-la-derivada-de-una-resta\"><\/span> Exemplos de derivada de subtra\u00e7\u00e3o<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Depois de vermos qual \u00e9 a f\u00f3rmula da derivada de uma subtra\u00e7\u00e3o, passamos agora \u00e0 an\u00e1lise de v\u00e1rios exemplos de derivadas deste tipo de opera\u00e7\u00f5es para compreender completamente como s\u00e3o derivadas as subtra\u00e7\u00f5es de fun\u00e7\u00f5es. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-1-derivada-de-una-resta-de-funciones-potenciales\"><\/span> Exemplo 1: Derivada de uma subtra\u00e7\u00e3o de fun\u00e7\u00f5es potenciais<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ea1b9e305c4c2b69fa19c4458715aa03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^3-4x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"117\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> A derivada da subtra\u00e7\u00e3o de duas fun\u00e7\u00f5es \u00e9 equivalente \u00e0 diferen\u00e7a das derivadas de cada fun\u00e7\u00e3o separadamente. Portanto, primeiro calcularemos a derivada de cada fun\u00e7\u00e3o separadamente: <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-57\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d80e76a954f8534d7a4229864ab4c7ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{d}{dx} \\ x^3=3x^2\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"93\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c96360191da869602a245eff5e086e86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{d}{dx}\\ 4x=4\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"77\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p> A derivada de toda a fun\u00e7\u00e3o \u00e9, portanto, a seguinte: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e0227a82cee066220095b0cb8a73276f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=3x^2-4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"120\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-2-derivada-de-una-resta-de-funciones-distintas\"><\/span> Exemplo 2: Derivada de uma subtra\u00e7\u00e3o de diferentes fun\u00e7\u00f5es<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3359936db8c2787dcb9d1c9eb1afb194_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cos}(x)-\\log_7(x^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"189\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Para diferenciar fun\u00e7\u00f5es de subtra\u00e7\u00e3o, primeiro voc\u00ea deve diferenciar as duas fun\u00e7\u00f5es separadamente e depois subtra\u00ed-las. <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-60\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-88911e0c596da5c7e5cd27ae8411eda1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{d}{dx} \\ \\text{cos}(x)=-\\text{sen}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"158\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-440c15ed64920bf34bb54bc46cece0e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{d}{dx}\\ \\log_7 (x^2)=\\cfrac{2x}{x^2\\cdot \\ln(7)}=\\cfrac{2}{x\\ln(7)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"261\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p> E depois de fazer as duas derivadas, subtra\u00edmo-las com a mesma ordem inicial: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2e37e34506772fb550148f02cbc9ebdd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=-\\text{sen}(x)-\\cfrac{2}{x\\ln(7)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"200\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-3-derivada-de-una-resta-al-cuadrado\"><\/span> Exemplo 3: Derivada de uma subtra\u00e7\u00e3o quadrada<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3fea677e30feb8612d8d9ff7108a1625_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\left(x^7-2x^3-9\\right)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"178\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p> Neste caso temos uma fun\u00e7\u00e3o composta, pois \u00e9 uma subtra\u00e7\u00e3o entre tr\u00eas fun\u00e7\u00f5es ao quadrado. Devemos, portanto, usar a f\u00f3rmula da derivada de uma fun\u00e7\u00e3o potencial e a regra da cadeia para calcular a derivada de toda a fun\u00e7\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4592b6df0dc2ed15a07f4382a817372f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=2\\left(x^7-2x^3-9\\right)\\cdot \\left(7x^6-6x^2\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"285\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Veja:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/derivada-de-uma-funcao-potencial-de-potencia\/\">f\u00f3rmula para a derivada de uma pot\u00eancia<\/a><\/span> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-la-derivada-de-una-resta\"><\/span> Exerc\u00edcios resolvidos sobre a derivada de uma subtra\u00e7\u00e3o<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Derive as seguintes subtra\u00e7\u00f5es de fun\u00e7\u00f5es: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-554714ddbd627e6e4db47875cf74ced0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A) } f(x)=9x^3-5x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"152\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a44e7914b191221833e66b56cfb85c0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B) } f(x)=4x^5-6x^4-x^2-4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"229\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d2ffe7ee4f97af710c28c149b75ee6fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C) } f(x)=\\left(-3x^7-2x^5\\right)^4\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"196\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e2a65669468838e35b6314f216c94c20_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D) } f(x)=\\ln(5x^2+3x)-\\text{cos}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"249\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b2a845babbdb5c1108fb183266b265cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{E) } f(x)=8x^3-e^{5x}-\\sqrt{8x+2}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"241\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong> <\/div>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7619c845d231badc6717536370242c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A) } f'(x)=27x^2-5\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"154\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-69d4ff912fd82dfe90ff0996b1782195_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B) } f'(x)=20x^4-24x^3-2x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"221\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-efabd300a93704ba1ef8dd75ab565c32_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C) } f'(x)=4\\left(-3x^7-2x^5\\right)^3\\cdot (-21x^6-10x^4)\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"345\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ef722845586f56e6fa7aff817614b6b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D) } f'(x)=\\cfrac{10x+3}{5x^2+3x}-\\bilg(-\\text{sen}(x)\\bigr)=\\cfrac{10x+3}{5x^2+3x}+\\text{sen}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"420\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4cdf075f9332987c97d80e87aa96be07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{E) } f'(x)=24x^2-5e^{5x}-\\cfrac{8}{2\\sqrt{8x+2}}=24x^2-5e^{5x}-\\cfrac{4}{\\sqrt{8x+2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"476\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"demostracion-de-la-derivada-de-una-resta\"><\/span> Prova da derivada de uma subtra\u00e7\u00e3o<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A seguir, demonstraremos a f\u00f3rmula da derivada de uma subtra\u00e7\u00e3o de fun\u00e7\u00f5es da defini\u00e7\u00e3o da derivada, que \u00e9:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dc1699622d128f888c1f20599aeccf60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=\\lim_{h \\to 0}\\frac{f(x+h)-f(x)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"219\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> Ent\u00e3o, se z \u00e9 a diferen\u00e7a de duas fun\u00e7\u00f5es diferentes:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d195ec32e5caf166e1ae253669f8cea5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z(x)=f(x)-g(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"145\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-17f0dc216847872551637f676d1ff0b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle z'(x)=\\lim_{h \\to 0}\\frac{z(x+h)-z(x)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"215\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> Substitu\u00edmos z pela subtra\u00e7\u00e3o das fun\u00e7\u00f5es na express\u00e3o limite:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-40e01069bb5a50a9f8706e0b232a6c4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle z'(x)=\\lim_{h \\to 0}\\frac{\\bigl[f(x+h)-g(x+h)\\bigr]-\\bigl[f(x)-g(x)\\bigr]}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"389\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0c008a2b368b6662d4184f781c60b73d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle z'(x)=\\lim_{h \\to 0}\\frac{f(x+h)-g(x+h)-f(x)+g(x)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"359\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> Faremos agora uma transforma\u00e7\u00e3o para separar a fra\u00e7\u00e3o e obter a subtra\u00e7\u00e3o de duas fra\u00e7\u00f5es:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e88137498b1755e2cbe6a93a99d2b769_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle z'(x)=\\lim_{h \\to 0}\\left[\\frac{f(x+h)-f(x)}{h}+\\frac{-g(x+h)+g(x)}{h}\\right]\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"392\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4d2339c16c7467a19eb2387b7d84985e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle z'(x)=\\lim_{h \\to 0}\\left[\\frac{f(x+h)-f(x)}{h}-\\frac{g(x+h)-g(x)}{h}\\right]\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"379\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> Ao aplicar as leis dos limites, podemos separar a express\u00e3o acima em dois limites diferentes. Porque o limite de uma subtra\u00e7\u00e3o \u00e9 igual \u00e0 subtra\u00e7\u00e3o dos limites:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bd3b4a827d5e04a2c084d0b2cfc0a1af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle z'(x)=\\lim_{h \\to 0}\\frac{f(x+h)-f(x)}{h}-\\lim_{h \\to 0}\\frac{g(x+h)-g(x)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"394\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> Se voc\u00ea olhar de perto, cada limite corresponde \u00e0 derivada de uma fun\u00e7\u00e3o, o que significa que a f\u00f3rmula para a derivada de uma diferen\u00e7a \u00e9 satisfeita:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5e0eb0254f80f131053caa8ee1480b2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle z'(x)=f'(x)-g'(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"158\" style=\"vertical-align: -5px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Neste artigo explicamos como derivar uma subtra\u00e7\u00e3o de fun\u00e7\u00f5es (f\u00f3rmula). Voc\u00ea tamb\u00e9m encontrar\u00e1 exemplos de derivadas de subtra\u00e7\u00e3o e exerc\u00edcios resolvidos passo a passo para praticar. F\u00f3rmula para a derivada de uma subtra\u00e7\u00e3o A derivada da subtra\u00e7\u00e3o de duas fun\u00e7\u00f5es \u00e9 igual a subtrair a derivada de cada fun\u00e7\u00e3o separadamente. Por outras palavras, diferenciar duas &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/derivada-de-uma-subtracao\/\"> <span class=\"screen-reader-text\">Derivada de uma subtra\u00e7\u00e3o<\/span> Leia mais &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[11],"tags":[],"class_list":["post-34","post","type-post","status-publish","format-standard","hentry","category-derivados"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u25b7 Derivada de uma subtra\u00e7\u00e3o (f\u00f3rmula e exerc\u00edcios resolvidos)<\/title>\n<meta name=\"description\" content=\"Explicamos como derivar uma subtra\u00e7\u00e3o de fun\u00e7\u00f5es (f\u00f3rmula). \u2705 Com exerc\u00edcios resolvidos passo a passo sobre derivadas de subtra\u00e7\u00f5es. \u2705\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/pt\/derivada-de-uma-subtracao\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u25b7 Derivada de uma subtra\u00e7\u00e3o (f\u00f3rmula e exerc\u00edcios resolvidos)\" \/>\n<meta property=\"og:description\" content=\"Explicamos como derivar uma subtra\u00e7\u00e3o de fun\u00e7\u00f5es (f\u00f3rmula). \u2705 Com exerc\u00edcios resolvidos passo a passo sobre derivadas de subtra\u00e7\u00f5es. \u2705\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/pt\/derivada-de-uma-subtracao\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-09-17T11:01:40+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2eb280531ada831cf451f6440ac3cb7d_l3.png\" \/>\n<meta name=\"author\" content=\"Equipe Mathoridade\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"Equipe Mathoridade\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. tempo de leitura\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/pt\/derivada-de-uma-subtracao\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/derivada-de-uma-subtracao\/\"},\"author\":{\"name\":\"Equipe Mathoridade\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\"},\"headline\":\"Derivada de uma subtra\u00e7\u00e3o\",\"datePublished\":\"2023-09-17T11:01:40+00:00\",\"dateModified\":\"2023-09-17T11:01:40+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/pt\/derivada-de-uma-subtracao\/\"},\"wordCount\":513,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"articleSection\":[\"Derivados\"],\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/pt\/derivada-de-uma-subtracao\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/pt\/derivada-de-uma-subtracao\/\",\"url\":\"https:\/\/mathority.org\/pt\/derivada-de-uma-subtracao\/\",\"name\":\"\u25b7 Derivada de uma subtra\u00e7\u00e3o (f\u00f3rmula e exerc\u00edcios resolvidos)\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/#website\"},\"datePublished\":\"2023-09-17T11:01:40+00:00\",\"dateModified\":\"2023-09-17T11:01:40+00:00\",\"description\":\"Explicamos como derivar uma subtra\u00e7\u00e3o de fun\u00e7\u00f5es (f\u00f3rmula). \u2705 Com exerc\u00edcios resolvidos passo a passo sobre derivadas de subtra\u00e7\u00f5es. \u2705\",\"breadcrumb\":{\"@id\":\"https:\/\/mathority.org\/pt\/derivada-de-uma-subtracao\/#breadcrumb\"},\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathority.org\/pt\/derivada-de-uma-subtracao\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathority.org\/pt\/derivada-de-uma-subtracao\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathority.org\/pt\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Derivada de uma subtra\u00e7\u00e3o\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathority.org\/pt\/#website\",\"url\":\"https:\/\/mathority.org\/pt\/\",\"name\":\"Mathority\",\"description\":\"Onde a curiosidade encontra o c\u00e1lculo!\",\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathority.org\/pt\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"pt-BR\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/mathority.org\/pt\/#organization\",\"name\":\"Mathority\",\"url\":\"https:\/\/mathority.org\/pt\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"pt-BR\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png\",\"contentUrl\":\"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png\",\"width\":703,\"height\":151,\"caption\":\"Mathority\"},\"image\":{\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/\"}},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\",\"name\":\"Equipe Mathoridade\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"pt-BR\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"caption\":\"Equipe Mathoridade\"},\"sameAs\":[\"http:\/\/mathority.org\/pt\"]}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"\u25b7 Derivada de uma subtra\u00e7\u00e3o (f\u00f3rmula e exerc\u00edcios resolvidos)","description":"Explicamos como derivar uma subtra\u00e7\u00e3o de fun\u00e7\u00f5es (f\u00f3rmula). \u2705 Com exerc\u00edcios resolvidos passo a passo sobre derivadas de subtra\u00e7\u00f5es. \u2705","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathority.org\/pt\/derivada-de-uma-subtracao\/","og_locale":"pt_BR","og_type":"article","og_title":"\u25b7 Derivada de uma subtra\u00e7\u00e3o (f\u00f3rmula e exerc\u00edcios resolvidos)","og_description":"Explicamos como derivar uma subtra\u00e7\u00e3o de fun\u00e7\u00f5es (f\u00f3rmula). \u2705 Com exerc\u00edcios resolvidos passo a passo sobre derivadas de subtra\u00e7\u00f5es. \u2705","og_url":"https:\/\/mathority.org\/pt\/derivada-de-uma-subtracao\/","article_published_time":"2023-09-17T11:01:40+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2eb280531ada831cf451f6440ac3cb7d_l3.png"}],"author":"Equipe Mathoridade","twitter_card":"summary_large_image","twitter_misc":{"Escrito por":"Equipe Mathoridade","Est. tempo de leitura":"3 minutos"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathority.org\/pt\/derivada-de-uma-subtracao\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/pt\/derivada-de-uma-subtracao\/"},"author":{"name":"Equipe Mathoridade","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00"},"headline":"Derivada de uma subtra\u00e7\u00e3o","datePublished":"2023-09-17T11:01:40+00:00","dateModified":"2023-09-17T11:01:40+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/pt\/derivada-de-uma-subtracao\/"},"wordCount":513,"commentCount":0,"publisher":{"@id":"https:\/\/mathority.org\/pt\/#organization"},"articleSection":["Derivados"],"inLanguage":"pt-BR","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathority.org\/pt\/derivada-de-uma-subtracao\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/pt\/derivada-de-uma-subtracao\/","url":"https:\/\/mathority.org\/pt\/derivada-de-uma-subtracao\/","name":"\u25b7 Derivada de uma subtra\u00e7\u00e3o (f\u00f3rmula e exerc\u00edcios resolvidos)","isPartOf":{"@id":"https:\/\/mathority.org\/pt\/#website"},"datePublished":"2023-09-17T11:01:40+00:00","dateModified":"2023-09-17T11:01:40+00:00","description":"Explicamos como derivar uma subtra\u00e7\u00e3o de fun\u00e7\u00f5es (f\u00f3rmula). \u2705 Com exerc\u00edcios resolvidos passo a passo sobre derivadas de subtra\u00e7\u00f5es. \u2705","breadcrumb":{"@id":"https:\/\/mathority.org\/pt\/derivada-de-uma-subtracao\/#breadcrumb"},"inLanguage":"pt-BR","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathority.org\/pt\/derivada-de-uma-subtracao\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathority.org\/pt\/derivada-de-uma-subtracao\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathority.org\/pt\/"},{"@type":"ListItem","position":2,"name":"Derivada de uma subtra\u00e7\u00e3o"}]},{"@type":"WebSite","@id":"https:\/\/mathority.org\/pt\/#website","url":"https:\/\/mathority.org\/pt\/","name":"Mathority","description":"Onde a curiosidade encontra o c\u00e1lculo!","publisher":{"@id":"https:\/\/mathority.org\/pt\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mathority.org\/pt\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"pt-BR"},{"@type":"Organization","@id":"https:\/\/mathority.org\/pt\/#organization","name":"Mathority","url":"https:\/\/mathority.org\/pt\/","logo":{"@type":"ImageObject","inLanguage":"pt-BR","@id":"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/","url":"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png","contentUrl":"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png","width":703,"height":151,"caption":"Mathority"},"image":{"@id":"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/"}},{"@type":"Person","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00","name":"Equipe Mathoridade","image":{"@type":"ImageObject","inLanguage":"pt-BR","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","caption":"Equipe Mathoridade"},"sameAs":["http:\/\/mathority.org\/pt"]}]}},"yoast_meta":{"yoast_wpseo_title":"","yoast_wpseo_metadesc":"","yoast_wpseo_canonical":""},"_links":{"self":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts\/34","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/comments?post=34"}],"version-history":[{"count":0,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts\/34\/revisions"}],"wp:attachment":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/media?parent=34"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/categories?post=34"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/tags?post=34"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}