{"id":154194,"date":"2022-03-25T23:15:52","date_gmt":"2022-03-25T17:45:52","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/differentiation-laws-rules-problems-and-faqs\/"},"modified":"2023-01-02T18:14:11","modified_gmt":"2023-01-02T12:44:11","slug":"differentiation-laws-rules-problems-and-faqs","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/differentiation-laws\/","title":{"rendered":"Differentiation Laws &#8211; Rules, Problems and FAQs"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/differentiation-laws\/#What_are_Derivatives\" title=\"What are Derivatives?\">What are Derivatives?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/differentiation-laws\/#1_Constant_Rule\" title=\"1) Constant Rule\">1) Constant Rule<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/differentiation-laws\/#2_Differentiation_of_a_Function_is_Linear\" title=\"2) Differentiation of a Function is Linear.\">2) Differentiation of a Function is Linear.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/differentiation-laws\/#3_Product_Rule\" title=\"3) Product Rule\">3) Product Rule<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/differentiation-laws\/#4_Chain_Rule\" title=\"4) Chain Rule\">4) Chain Rule<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/differentiation-laws\/#5_Polynomial_or_Elementary_Power_Rule\" title=\"5) Polynomial or Elementary Power Rule\">5) Polynomial or Elementary Power Rule<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/differentiation-laws\/#6_Quotient_Rule\" title=\"6) Quotient Rule\">6) Quotient Rule<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/differentiation-laws\/#7_The_Derivative_of_Trigonometric_Functions\" title=\"7) The Derivative of Trigonometric Functions\">7) The Derivative of Trigonometric Functions<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/differentiation-laws\/#8_The_Derivative_of_an_Exponential_Function\" title=\"8) The Derivative of an Exponential Function\">8) The Derivative of an Exponential Function<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/differentiation-laws\/#9_The_Derivative_of_Logarithmic_Functions\" title=\"9) The Derivative of Logarithmic Functions\">9) The Derivative of Logarithmic Functions<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/differentiation-laws\/#Problems_on_Calculus_Differentiation_Rules\" title=\"Problems on Calculus Differentiation Rules\">Problems on Calculus Differentiation Rules<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"What_are_Derivatives\"><\/span>What are Derivatives?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>A derivative is a security or financial instrument whose value is based on, or &#8220;derived&#8221; from, the value of an underlying asset, index, or reference rate. The underlying asset, index, or reference rate can be a single security, a basket of securities, an index, or a reference rate. Derivatives can be used for hedging (protecting against losses) or for speculation (taking advantage of price movements).<\/p>\n<p>The most common types of derivatives are futures, options, and swaps. Futures are contracts to buy or sell an underlying asset, index, or reference rate at a future date. Options are contracts that give the holder the right, but not the obligation, to buy or sell an underlying asset, index, or reference rate at a future date. Swaps are contracts to exchange one type of financial instrument for another at a future date.<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-154193 size-medium\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/differentiation-laws-rules-problems-and-faqs-300x212.jpg\" alt=\"Differentiation Laws \" width=\"300\" height=\"212\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/differentiation-laws-rules-problems-and-faqs-300x212.jpg?v=1648230346 300w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/differentiation-laws-rules-problems-and-faqs.jpg?v=1648230346 606w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"1_Constant_Rule\"><\/span>1) Constant Rule<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>: A constant rule is a rule that always produces the same output for a given input. For example, the rule &#8220;add 2 to every number&#8221; is a constant rule, because for any number x, the output will always be 2x. 2) Recursive Rule : A recursive rule is a rule that produces a new output based on the previous output. For example, the rule &#8220;multiply by 2 every time&#8221; is a recursive rule, because for any number x, the output will be 2&#215;2, then 2&#215;4, then 2&#215;8, and so on.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"2_Differentiation_of_a_Function_is_Linear\"><\/span>2) Differentiation of a Function is Linear.<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Differentiation of a function is linear if the derivative of the function is a linear function of the input variable. This means that the derivative of the function can be expressed as a linear combination of the derivatives of the individual input variables.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"3_Product_Rule\"><\/span>3) Product Rule<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The product rule states that the derivative of the product of two functions is the product of the derivatives of the individual functions.<\/p>\n<p>(4) Quotient Rule<\/p>\n<p>The quotient rule states that the derivative of the quotient of two functions is the derivative of the numerator minus the derivative of the denominator multiplied by the quotient of the two functions.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"4_Chain_Rule\"><\/span>4) Chain Rule<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The chain rule states that the derivative of the composite function of two functions is the derivative of the first function multiplied by the derivative of the second function.<\/p>\n<p>The chain rule can be represented using the following notation:<\/p>\n<p>where &#8220;f&#8221; and &#8220;g&#8221; are functions, and &#8220;d&#8221; is the derivative.<\/p>\n<p>The chain rule can be applied to functions of more than two variables.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"5_Polynomial_or_Elementary_Power_Rule\"><\/span>5) Polynomial or Elementary Power Rule<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>This rule states that the derivative of a polynomial is the sum of the derivatives of the individual terms in the polynomial. The derivative of an elementary power is the product of the derivative of the power and the coefficient of the power.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"6_Quotient_Rule\"><\/span>6) Quotient Rule<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The Quotient Rule states that:<\/p>\n<p>The derivative of the quotient of two functions is the derivative of the first function divided by the derivative of the second function.<\/p>\n<p>In symbols:<\/p>\n<p>where:<\/p>\n<p>is the derivative of<\/p>\n<p>Proof<\/p>\n<p>We will use the definition of the derivative to prove the quotient rule.<\/p>\n<p>We have:<\/p>\n<p>We also have:<\/p>\n<p>And so:<\/p>\n<h2><span class=\"ez-toc-section\" id=\"7_The_Derivative_of_Trigonometric_Functions\"><\/span>7) The Derivative of Trigonometric Functions<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The derivative of a trigonometric function is a function that is the derivative of the original function. The derivative of a trigonometric function is found by taking the derivative of the function and then applying the chain rule.<\/p>\n<p>The derivative of a trigonometric function can be used to find the instantaneous rate of change of the function at a certain point in time. This can be helpful in solving problems that require calculus.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"8_The_Derivative_of_an_Exponential_Function\"><\/span>8) The Derivative of an Exponential Function<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The derivative of an exponential function is given by the following equation:<\/p>\n<p>The derivative of an exponential function is equal to the product of the derivative of the base and the derivative of the exponent.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"9_The_Derivative_of_Logarithmic_Functions\"><\/span>9) The Derivative of Logarithmic Functions<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The derivative of the logarithmic function can be found by taking the derivative of the natural logarithm function.<\/p>\n<p>The derivative of the natural logarithm function is:<\/p>\n<p>The derivative of the logarithmic function is:<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Problems_on_Calculus_Differentiation_Rules\"><\/span>Problems on Calculus Differentiation Rules<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>1. Find the derivative of \\begin{align*}f(x) = x^3\\end{align*}<\/p>\n<p>The derivative of \\begin{align*}x^3\\end{align*} is \\begin{align*}3x^2\\end{align*}.<\/p>\n<p>2. Find the derivative of \\begin{align*}f(x) = 3x^2\\end{align*}<\/p>\n<p>The derivative of \\begin{align*}3x^2\\end{align*} is \\begin{align*}6x\\end{align*}.<\/p>\n<p>3. Find the derivative of \\begin{align*}f(x) = x^5\\end{align*}<\/p>\n<p>The derivative of \\begin{align*}x^5\\end{align*} is \\begin{align*}5x^4\\end{align*}.<\/p>\n<p>4. Find the derivative of \\begin{align*}f(x) = sin(x)\\end{align*}<\/p>\n<p>The derivative of \\begin{align*}sin(x)\\end{align*} is \\begin{align*}cos(x)\\end{align*}.<\/p>\n<p>5. Find the derivative of \\begin{align*}f(x) = x^4\\end{align*}<\/p>\n<p>The derivative of \\begin{align*}x^<\/p>\n","protected":false},"excerpt":{"rendered":"<p>What are Derivatives? A derivative is a security or financial instrument whose value is based on, or &#8220;derived&#8221; from, the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Differentiation Laws","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"Learn about differentiation laws topic of Maths in details explained by subject experts on infinitylearn.com.","custom_permalink":"maths\/differentiation-laws\/"},"categories":[13],"tags":[],"table_tags":[],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Differentiation Laws - Rules, Problems and FAQs - Infinity Learn by Sri Chaitanya<\/title>\n<meta name=\"description\" content=\"Learn about differentiation laws topic of Maths in details explained by subject experts on infinitylearn.com.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/differentiation-laws\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Differentiation Laws - 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