{"id":10170,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=10170"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"implicit-differentiations","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/implicit-differentiations\/","title":{"rendered":"Implicit Differentiations"},"content":{"rendered":"<div class=\"article-watermark-wrapper\">\n<div style=\"position: relative; z-index: 1;\">\n<p style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 9pt; color: #444444;\">KAPDEC&reg; | Elite STEM Learning Platform | <a href=\"https:\/\/kapdec.com\" target=\"_blank\" rel=\"noopener noreferrer\" style=\"color: #444444; text-decoration: underline;\">https:\/\/kapdec.com<\/a><\/p>\n<hr \/>\n<h2><strong>Unit: <\/strong><strong>Differentiation-Composite, Implicit &amp; Inverse function<\/strong><\/h2>\n<h3><strong>Chapter: <\/strong><strong>Implicit Differentiation<\/strong><\/h3>\n<p><em>Reference: &#8211; Implicit Functions, Implicit equations, Derivatives of Implicit relations, Chain rules, Slope of tangent lines, Equation of tangent line, Implicit differentiation, Higher order derivatives, vertical &amp; Horizontal Tangent lines, Curve sketching, Optimization, Trigonometric Functions.<\/em><\/p>\n<p><strong>After studying this chapter, you should be able to:<\/strong><\/p>\n<ul>\n<li>Differentiation Rule &amp; Finding Derivatives.<\/li>\n<li>High-order Derivatives.<\/li>\n<li>Tangent &amp; Normal lines, Related rates.<\/li>\n<li>Second derivatives &amp; Concavity<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<p><strong><u>Introduction to Implicit Differentiation<\/u><\/strong><\/p>\n<ol>\n<li><u>Definition of Implicit Differentiation<\/u>:<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<ul>\n<li>Implicit differentiation is a technique used to find the derivative of an implicitly defined function.<\/li>\n<\/ul>\n<ul>\n<li>An implicitly defined function is given in the form of an equation that may involve multiple variables.<\/li>\n<li>Unlike explicit functions, which are solved explicitly for a dependent variable, implicit functions do not isolate the dependent variable.<\/li>\n<\/ul>\n<ol>\n<li><u>Differentiation Rules<\/u>:<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<ul>\n<li>Implicit differentiation involves applying the basic rules of differentiation to both sides of the equation.<\/li>\n<li>These rules include the power rule, product rule, quotient rule, and chain rule.<\/li>\n<li>When differentiating for a particular variable, all other variables are treated as constants.<\/li>\n<\/ul>\n<ol>\n<li><u>Finding Derivatives of Implicit Equations<\/u>:<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<ul>\n<li>To find the derivative of an implicitly defined function, you differentiate both sides of the equation for the independent variable.<\/li>\n<li>When differentiating terms that involve the dependent variable, you apply the chain rule.<\/li>\n<li>The derivative of the dependent variable is represented using the notation dy\/dx.<\/li>\n<\/ul>\n<ol>\n<li><u>Higher-Order Derivatives<\/u>:<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<ul>\n<li>Implicit differentiation can be extended to find higher-order derivatives of implicitly defined functions.<\/li>\n<li>To find the second derivative, you differentiate the first derivative obtained through implicit differentiation.<\/li>\n<\/ul>\n<ol>\n<li><u>Tangent Lines and Normal Lines<\/u>:<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<ul>\n<li>Implicit differentiation allows you to find the equations of tangent lines and normal lines to curves defined implicitly.<\/li>\n<li>To find the slope of the tangent line at a given point, you substitute the coordinates of the point into the derivative expression obtained through implicit differentiation.<\/li>\n<li>The slope of the tangent line is used to find the equation of the tangent line using point-slope form.<\/li>\n<li>The normal line is perpendicular to the tangent line and has a slope that is the negative reciprocal of the tangent line&#8217;s slope.<\/li>\n<\/ul>\n<ol>\n<li><u>Related Rates<\/u>:<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<ul>\n<li>Implicit differentiation is useful for solving related rate problems.<\/li>\n<li>Related rates problems involve finding the rate at which one variable changes for another variable.<\/li>\n<li>Implicit differentiation allows you to differentiate both sides of an equation for time (or another variable) and solve for the desired rate.<\/li>\n<\/ul>\n<ol>\n<li><u>Second Derivative and Concavity<\/u>:<\/li>\n<\/ol>\n<ul>\n<li>Implicit differentiation can be used to find the second derivative of implicitly defined functions.<\/li>\n<li>The second derivative provides information about the concavity and points of inflection of the curve defined implicitly.<\/li>\n<\/ul>\n<ol>\n<li><u>Optimization<\/u>:<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<ul>\n<li>Implicit differentiation can be applied to optimization problems, where you need to find the maximum or minimum values of a quantity given certain constraints.<\/li>\n<li>By setting the derivative of the implicitly defined function equal to zero and solving for the independent variable, you can find critical points that correspond to extreme values.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<p><strong><u>Differentiation Rule &amp; Finding Derivatives:<\/u><\/strong><\/p>\n<p>\u00a0<\/p>\n<ol>\n<li><u>Power Rule<\/u>:\n<ul>\n<li>If you have a function of the form f(x) = x<sup>n<\/sup>, where n is a constant, the derivative is given by f'(x) = nx<sup>(n-1)<\/sup>.<\/li>\n<li>For example, if f(x) = x<sup>2<\/sup>, the derivative is f'(x) = 2x.<\/li>\n<\/ul>\n<\/li>\n<li><u>Product Rule<\/u>:\n<ul>\n<li>If you have two functions, u(x) and v(x), the derivative of their product is given by (u(x)v(x))&#8217; = u'(x)v(x) + u(x)v'(x).<\/li>\n<li>For example, if f(x) = x<sup>2<\/sup> * sin(x), then f'(x) = (2x * sin(x)) + (x<sup>2<\/sup> * cos(x)).<\/li>\n<\/ul>\n<\/li>\n<li><u>Quotient Rule<\/u>:\n<ul>\n<li>If you have two functions, u(x) and v(x), the derivative of their quotient is given by (u(x)\/v(x))&#8217; = (u'(x)v(x) &#8211; u(x)v'(x))\/(v(x))<sup>2<\/sup>.<\/li>\n<li>For example, if f(x) = (x<sup>2<\/sup> + 1) \/ x, then f'(x) = ((2x * x) &#8211; (x<sup>2<\/sup> + 1))\/(x<sup>2<\/sup>)<sup>2<\/sup>.<\/li>\n<\/ul>\n<\/li>\n<li><u>Chain Rule<\/u>:\n<ul>\n<li>The chain rule is used when you have a composition of functions.<\/li>\n<li>If you have a function y = f(g(x)), where g(x) is an inner function and f(u) is an outer function, the derivative is given by dy\/dx = f'(g(x)) * g'(x).<\/li>\n<li>For example, if f(x) = sin(2x), then f'(x) = 2 * cos(2x).<\/li>\n<\/ul>\n<\/li>\n<li><u>Trigonometric Functions<\/u>:\n<ul>\n<li>The derivatives of common trigonometric functions are:\n<ul>\n<li>d\/dx (sin(x)) = cos(x)<\/li>\n<li>d\/dx (cos(x)) = -sin(x)<\/li>\n<li>d\/dx (tan(x)) = sec<sup>2<\/sup>(x)<\/li>\n<\/ul>\n<\/li>\n<li>Similarly, you can find the derivatives of other trigonometric functions using the chain rule and the quotient rule.<\/li>\n<\/ul>\n<\/li>\n<li><u>Exponential and Logarithmic Functions<\/u>:\n<ul>\n<li>The derivatives of exponential and logarithmic functions are:\n<ul>\n<li>d\/dx (e<sup>x<\/sup>) = e<sup>x<\/sup><\/li>\n<li>d\/dx (ln(x)) = <em>1x<\/em>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"36\" src=\"https:\/\/app.kapdec.com\/questions-images\/zfPJAjSuudu71735818124.png?time=1735818124\" width=\"9\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<\/li>\n<\/ul>\n<\/li>\n<li>For other exponential and logarithmic functions, you can use the chain rule to find their derivatives.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<p>\u00a0<\/p>\n<p><strong><u>Tangent, Normal lines &amp; Related rates<\/u><\/strong><strong>: &#8211;<\/strong><\/p>\n<p><strong><u>Implicit Differentiation &amp; High Order Derivatives<\/u><\/strong>: &#8211;<\/p>\n<p>\u00a0<\/p>\n<ol>\n<li><u>Implicit Functions<\/u>: An implicit function is a function where the dependent and independent variables are not explicitly separated. In other words, the equation defining the function does not express y explicitly in terms of x.<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<ol>\n<li><u>Implicit Differentiation<\/u>: Implicit differentiation is a technique used to find the derivatives of implicitly defined functions.<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<ol>\n<li><u>Procedure<\/u>: To perform implicit differentiation, you treat the dependent variable y as a function of x and differentiate both sides of the equation for x. However, when differentiating y for x, you also need to consider the chain rule since y is not explicitly expressed in terms of x.<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<ol>\n<li><u>Chain Rule in Implicit Differentiation<\/u>: When differentiating y for x, you apply the chain rule to the terms involving y. This involves multiplying the derivative of y for x (dy\/dx) by the derivative of the term inside the brackets for y.<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<ol>\n<li><u>Simplification<\/u>: After applying implicit differentiation, you can simplify the resulting equation to solve for dy\/dx, which represents the derivative of the implicitly defined function.<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<ol>\n<li><u>Applications<\/u>: Implicit differentiation is commonly used in various mathematical fields, such as physics and engineering, where relationships between variables are defined implicitly.<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<p><strong>Example<\/strong>: &#8211; Show that the differential equation (x \u2013 y) <em>dy\/dx<\/em> \u00a0= x + 2y is homogeneous and solve it.<\/p>\n<p><strong>Solution:<\/strong> (x \u2013 y) <em>dy\/dx<\/em> \u00a0= x + 2y<\/p>\n<p><div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"39\" src=\"https:\/\/app.kapdec.com\/questions-images\/f2fzVBtkGvBy1735818124.png?time=1735818125\" width=\"81\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 &#8230; (1)<\/p>\n<p>Let F(x, y) = <\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"38\" src=\"https:\/\/app.kapdec.com\/questions-images\/PQI7eFSWm0QE1735818124.png?time=1735818125\" width=\"37\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>Now,<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"40\" src=\"https:\/\/app.kapdec.com\/questions-images\/WxOfe8rDXYRc1735818124.png?time=1735818125\" width=\"168\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>Therefore, F(x, y) is a homogenous function of degree zero. So, the given differential equation is a homogenous differential equation.<\/p>\n<p>Put y = vx \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 &#8230; (2)<\/p>\n<p>Differentiating equation (2) with respect to, x we get<\/p>\n<p><div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"36\" src=\"https:\/\/app.kapdec.com\/questions-images\/R4WNghMhNR271735818125.png?time=1735818125\" width=\"112\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 &#8230; (3)<\/p>\n<p>Substituting the value of y and <em>dy\/dx<\/em> \u00a0in equation (1) we get<\/p>\n<p><div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"36\" src=\"https:\/\/app.kapdec.com\/questions-images\/xmFTWUp2j5DB1735818125.png?time=1735818126\" width=\"129\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0<\/p>\n<p>Integrating both sides of equation (5), we get<\/p>\n<p><div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"36\" src=\"https:\/\/app.kapdec.com\/questions-images\/8pBUUpj0Zba91735818126.png?time=1735818126\" width=\"172\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0<\/p>\n<p><div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"36\" src=\"https:\/\/app.kapdec.com\/questions-images\/oxQegta3p5LW1735818126.png?time=1735818127\" width=\"244\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0<\/p>\n<p><div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"36\" src=\"https:\/\/app.kapdec.com\/questions-images\/jx9mKvRZ9fj11735818126.png?time=1735818127\" width=\"374\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0<\/p>\n<p><div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"54\" src=\"https:\/\/app.kapdec.com\/questions-images\/9YaZche97bqW1735818126.png?time=1735818127\" width=\"453\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p> \u00a0<\/p>\n<p>\u00a0<\/p>\n<p>which is the general solution of the given differential equation.<\/p>\n<p><strong><u>Key Points<\/u><\/strong><\/p>\n<ul>\n<li>The chain rule is fundamental in calculus and used to differentiate composite functions.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>It provides a method to find the derivative of a function that is composed of multiple functions.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>The chain rule states that if y = f(g(x)), then dy\/dx = f'(g(x)) * g'(x).<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>f'(g(x)) represents the derivative of the outer function f(u) evaluated at the inner function g(x).<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>g'(x) represents the derivative of the inner function g(x) for x.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>The chain rule allows us to break down complex functions into simpler functions and differentiate them separately.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>It is essential to apply the chain rule when the function involves compositions of functions like exponentials, logarithms, or trigonometric functions.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>The chain rule can be extended to functions with multiple variables using partial derivatives.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Implicit differentiation uses the chain rule to find derivatives of implicitly defined functions.<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<ul>\n<li>Understanding and correctly applying the chain rule is crucial for solving differentiation problems and has various applications in mathematics, science, and engineering.<\/li>\n<\/ul>\n<p><!--kapdec-footer-start--><\/p>\n<style>.kapdec-article-footer{font-family:Arial,Helvetica,Calibri,sans-serif;color:#444;}.kapdec-footer-grid{display:flex;align-items:stretch;border:1px solid #e5e7eb;border-radius:6px;overflow:hidden;}.kapdec-footer-left,.kapdec-qr-block{flex:1 1 50%;width:50%;box-sizing:border-box;min-width:0;}.kapdec-footer-left{padding:22px 28px;border-right:1px solid #e5e7eb;}.kapdec-citation-block{line-height:1.6;font-size:9pt;color:#333;margin:0;}.kapdec-citation-block p{margin:0 0 10px 0;}.kapdec-citation-block a{color:#0066cc;text-decoration:underline;}.kapdec-copyright-block{margin-top:18px;padding-top:14px;border-top:1px solid #e5e7eb;font-size:7.5pt;color:#777;line-height:1.55;text-align:left;}.kapdec-copyright-block p{margin:0 0 5px 0;}.kapdec-qr-block{padding:22px 28px;display:flex;flex-direction:column;align-items:center;justify-content:center;text-align:center;}.kapdec-qr-label{margin:0 0 8px 0;font-size:8.5pt;font-weight:600;color:#444;line-height:1.35;letter-spacing:.02em;}.kapdec-qr-url{margin:0 0 14px 0;font-size:7.5pt;line-height:1.4;color:#777;word-break:break-word;max-width:100%;}.kapdec-qr-url a{color:#777;text-decoration:underline;}@media (max-width:640px){.kapdec-footer-grid{flex-direction:column;}.kapdec-footer-left,.kapdec-qr-block{width:100%;flex-basis:100%;border-right:none;}.kapdec-footer-left{border-bottom:1px solid #e5e7eb;}}<\/style>\n<div class=\"kapdec-article-footer\" style=\"margin-top: 28px; 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