{"id":10180,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=10180"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"implicit-differentiation","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/implicit-differentiation\/","title":{"rendered":"Implicit Differentiation"},"content":{"rendered":"
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KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

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Unit: <\/strong>Differentiation-Composite, Implicit & Inverse function<\/strong><\/h2>\n

Chapter: <\/strong>Implicit Differentiation<\/strong><\/h3>\n

Reference: – Implicit Functions, Implicit equations, Derivatives of Implicit relations, Chain rules, Slope of tangent lines, Equation of tangent line, Implicit differentiation, Higher order derivatives, vertical & Horizontal Tangent lines, Curve sketching, Optimization, Trigonometric Functions.<\/em><\/p>\n

After studying this chapter, you should be able to:<\/strong><\/p>\n

    \n
  • Differentiation Rule & Finding Derivatives.<\/li>\n
  • High-order Derivatives.<\/li>\n
  • Tangent & Normal lines, Related rates.<\/li>\n
  • Second derivatives & Concavity<\/li>\n<\/ul>\n

    \u00a0<\/p>\n

    Introduction to Implicit Differentiation<\/u><\/strong><\/p>\n

      \n
    1. Definition of Implicit Differentiation<\/u>:<\/li>\n<\/ol>\n

      \u00a0<\/p>\n

        \n
      • Implicit differentiation is a technique used to find the derivative of an implicitly defined function.<\/li>\n<\/ul>\n
          \n
        • An implicitly defined function is given in the form of an equation that may involve multiple variables.<\/li>\n
        • Unlike explicit functions, which are solved explicitly for a dependent variable, implicit functions do not isolate the dependent variable.<\/li>\n<\/ul>\n
            \n
          1. Differentiation Rules<\/u>:<\/li>\n<\/ol>\n

            \u00a0<\/p>\n

              \n
            • Implicit differentiation involves applying the basic rules of differentiation to both sides of the equation.<\/li>\n
            • These rules include the power rule, product rule, quotient rule, and chain rule.<\/li>\n
            • When differentiating for a particular variable, all other variables are treated as constants.<\/li>\n<\/ul>\n
                \n
              1. Finding Derivatives of Implicit Equations<\/u>:<\/li>\n<\/ol>\n

                \u00a0<\/p>\n

                  \n
                • To find the derivative of an implicitly defined function, you differentiate both sides of the equation for the independent variable.<\/li>\n
                • When differentiating terms that involve the dependent variable, you apply the chain rule.<\/li>\n
                • The derivative of the dependent variable is represented using the notation dy\/dx.<\/li>\n<\/ul>\n
                    \n
                  1. Higher-Order Derivatives<\/u>:<\/li>\n<\/ol>\n

                    \u00a0<\/p>\n

                      \n
                    • Implicit differentiation can be extended to find higher-order derivatives of implicitly defined functions.<\/li>\n
                    • To find the second derivative, you differentiate the first derivative obtained through implicit differentiation.<\/li>\n<\/ul>\n
                        \n
                      1. Tangent Lines and Normal Lines<\/u>:<\/li>\n<\/ol>\n

                        \u00a0<\/p>\n

                          \n
                        • Implicit differentiation allows you to find the equations of tangent lines and normal lines to curves defined implicitly.<\/li>\n
                        • 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
                        • The slope of the tangent line is used to find the equation of the tangent line using point-slope form.<\/li>\n
                        • The normal line is perpendicular to the tangent line and has a slope that is the negative reciprocal of the tangent line’s slope.<\/li>\n<\/ul>\n
                            \n
                          1. Related Rates<\/u>:<\/li>\n<\/ol>\n

                            \u00a0<\/p>\n

                              \n
                            • Implicit differentiation is useful for solving related rate problems.<\/li>\n
                            • Related rates problems involve finding the rate at which one variable changes for another variable.<\/li>\n
                            • 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
                                \n
                              1. Second Derivative and Concavity<\/u>:<\/li>\n<\/ol>\n
                                  \n
                                • Implicit differentiation can be used to find the second derivative of implicitly defined functions.<\/li>\n
                                • The second derivative provides information about the concavity and points of inflection of the curve defined implicitly.<\/li>\n<\/ul>\n
                                    \n
                                  1. Optimization<\/u>:<\/li>\n<\/ol>\n

                                    \u00a0<\/p>\n

                                      \n
                                    • 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
                                    • 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

                                      \u00a0<\/p>\n

                                      Differentiation Rule & Finding Derivatives:<\/u><\/strong><\/p>\n

                                      \u00a0<\/p>\n

                                        \n
                                      1. Power Rule<\/u>:\n
                                          \n
                                        • If you have a function of the form f(x) = xn<\/sup>, where n is a constant, the derivative is given by f'(x) = nx(n-1)<\/sup>.<\/li>\n
                                        • For example, if f(x) = x2<\/sup>, the derivative is f'(x) = 2x.<\/li>\n<\/ul>\n<\/li>\n
                                        • Product Rule<\/u>:\n
                                            \n
                                          • If you have two functions, u(x) and v(x), the derivative of their product is given by (u(x)v(x))’ = u'(x)v(x) + u(x)v'(x).<\/li>\n
                                          • For example, if f(x) = x2<\/sup> * sin(x), then f'(x) = (2x * sin(x)) + (x2<\/sup> * cos(x)).<\/li>\n<\/ul>\n<\/li>\n
                                          • Quotient Rule<\/u>:\n
                                              \n
                                            • If you have two functions, u(x) and v(x), the derivative of their quotient is given by (u(x)\/v(x))’ = (u'(x)v(x) – u(x)v'(x))\/(v(x))2<\/sup>.<\/li>\n
                                            • For example, if f(x) = (x2<\/sup> + 1) \/ x, then f'(x) = ((2x * x) – (x2<\/sup> + 1))\/(x2<\/sup>)2<\/sup>.<\/li>\n<\/ul>\n<\/li>\n
                                            • Chain Rule<\/u>:\n
                                                \n
                                              • The chain rule is used when you have a composition of functions.<\/li>\n
                                              • 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
                                              • For example, if f(x) = sin(2x), then f'(x) = 2 * cos(2x).<\/li>\n<\/ul>\n<\/li>\n
                                              • Trigonometric Functions<\/u>:\n
                                                  \n
                                                • The derivatives of common trigonometric functions are:\n
                                                    \n
                                                  • d\/dx (sin(x)) = cos(x)<\/li>\n
                                                  • d\/dx (cos(x)) = -sin(x)<\/li>\n
                                                  • d\/dx (tan(x)) = sec2<\/sup>(x)<\/li>\n<\/ul>\n<\/li>\n
                                                  • Similarly, you can find the derivatives of other trigonometric functions using the chain rule and the quotient rule.<\/li>\n<\/ul>\n<\/li>\n
                                                  • Exponential and Logarithmic Functions<\/u>:\n
                                                      \n
                                                    • The derivatives of exponential and logarithmic functions are:\n
                                                        \n
                                                      • d\/dx (ex<\/sup>) = ex<\/sup><\/li>\n
                                                      • d\/dx (ln(x)) = 1x<\/em>\n
                                                        \"\"<\/p>\n

                                                        Source: Kapdec.com<\/p>\n<\/div>\n<\/li>\n<\/ul>\n<\/li>\n

                                                      • For other exponential and logarithmic functions, you can use the chain rule to find their derivatives.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n

                                                        \u00a0<\/p>\n

                                                        \u00a0<\/p>\n

                                                        Tangent, Normal lines & Related rates<\/u><\/strong>: –<\/strong><\/p>\n

                                                        Implicit Differentiation & High Order Derivatives<\/u><\/strong>: –<\/p>\n

                                                        \u00a0<\/p>\n

                                                          \n
                                                        1. 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

                                                          \u00a0<\/p>\n

                                                            \n
                                                          1. Implicit Differentiation<\/u>: Implicit differentiation is a technique used to find the derivatives of implicitly defined functions.<\/li>\n<\/ol>\n

                                                            \u00a0<\/p>\n

                                                              \n
                                                            1. 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

                                                              \u00a0<\/p>\n

                                                                \n
                                                              1. 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

                                                                \u00a0<\/p>\n

                                                                  \n
                                                                1. 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

                                                                  \u00a0<\/p>\n

                                                                    \n
                                                                  1. 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

                                                                    \u00a0<\/p>\n

                                                                    Example<\/strong>: – Show that the differential equation (x \u2013 y) dy\/dx<\/em> \u00a0= x + 2y is homogeneous and solve it.<\/p>\n

                                                                    Solution:<\/strong> (x \u2013 y) dy\/dx<\/em> \u00a0= x + 2y<\/p>\n

                                                                    \"\"<\/p>\n

                                                                    Source: Kapdec.com<\/p>\n<\/div>\n

                                                                    \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 … (1)<\/p>\n

                                                                    Let F(x, y) = <\/p>\n

                                                                    \"\"<\/p>\n

                                                                    Source: Kapdec.com<\/p>\n<\/div>\n

                                                                    Now,<\/p>\n

                                                                    \"\"<\/p>\n

                                                                    Source: Kapdec.com<\/p>\n<\/div>\n

                                                                    Therefore, F(x, y) is a homogenous function of degree zero. So, the given differential equation is a homogenous differential equation.<\/p>\n

                                                                    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 … (2)<\/p>\n

                                                                    Differentiating equation (2) with respect to, x we get<\/p>\n

                                                                    \"\"<\/p>\n

                                                                    Source: Kapdec.com<\/p>\n<\/div>\n

                                                                    \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 … (3)<\/p>\n

                                                                    Substituting the value of y and dy\/dx<\/em> \u00a0in equation (1) we get<\/p>\n

                                                                    \"\"<\/p>\n

                                                                    Source: Kapdec.com<\/p>\n<\/div>\n

                                                                    \u00a0<\/p>\n

                                                                    Integrating both sides of equation (5), we get<\/p>\n

                                                                    v-1<\/em>v<\/em>2<\/em>+v+1<\/em>dv=-<\/em>dxx<\/em><\/p>\n

                                                                    \"\"<\/p>\n

                                                                    Source: Kapdec.com<\/p>\n<\/div>\n

                                                                    \u00a0<\/p>\n

                                                                    1<\/em>2<\/em>2v+1-3<\/em>v<\/em>2<\/em>+v+1<\/em>dv=-<\/em>logx<\/em>+<\/em>C1<\/em><\/p>\n

                                                                    \"\"<\/p>\n

                                                                    Source: Kapdec.com<\/p>\n<\/div>\n

                                                                    \u00a0<\/p>\n

                                                                    1<\/em>2<\/em>2v+1<\/em>v<\/em>2<\/em>+v+1<\/em>dv-<\/em>3<\/em>2<\/em>1<\/em>v<\/em>2<\/em>+v+1<\/em>dv=-<\/em>logx<\/em>+<\/em>C1<\/em><\/p>\n

                                                                    \"\"<\/p>\n

                                                                    Source: Kapdec.com<\/p>\n<\/div>\n

                                                                    \u00a0<\/p>\n

                                                                    1<\/em>2<\/em>log<\/em>v<\/em>2<\/em>+v+1<\/em>–<\/em>3<\/em>2<\/em>1<\/em>v+<\/em>1<\/em>2<\/em>2<\/em>+<\/em>3<\/em>2<\/em>2<\/em>dv=-<\/em>logx<\/em>+<\/em>C1<\/em><\/p>\n

                                                                    \"\"<\/p>\n

                                                                    Source: Kapdec.com<\/p>\n<\/div>\n

                                                                    \u00a0<\/p>\n

                                                                    1<\/em>2<\/em>log<\/em>v<\/em>2<\/em>+v+1<\/em>–<\/em>3<\/em>2<\/em>.<\/em>2<\/em>3<\/em>tan<\/em>-1<\/em>2v+1<\/em>3<\/em>=-<\/em>logx<\/em>+<\/em>C1<\/em> \u00a0<\/p>\n

                                                                    1<\/em>2<\/em>log<\/em>v<\/em>2<\/em>+v+1<\/em>+<\/em>1<\/em>2<\/em> <\/em>logx<\/em>2<\/em>=<\/em>3<\/em> <\/em>tan<\/em>-1<\/em>2v+1<\/em>3<\/em>+<\/em>C1<\/em> \u00a0<\/p>\n

                                                                    Replacing v by y\/x<\/em> \u00a0, we get<\/p>\n

                                                                    1<\/em>2<\/em>logy<\/em>2<\/em>x<\/em>2<\/em>+<\/em>y<\/em>x<\/em>+1<\/em>+<\/em>1<\/em>2<\/em> <\/em>logx<\/em>2<\/em>=<\/em>3<\/em> <\/em>tan<\/em>-1<\/em>2y+x<\/em>3<\/em>x<\/em>+<\/em>C1<\/em> \u00a0<\/p>\n

                                                                    1<\/em>2<\/em>logy<\/em>2<\/em>x<\/em>2<\/em>+<\/em>y<\/em>x<\/em>+1<\/em>x<\/em>2<\/em>=<\/em>3<\/em> <\/em>tan<\/em>-1<\/em>2y+x<\/em>3<\/em>x<\/em>+<\/em>C1<\/em> \u00a0<\/p>\n

                                                                    log y<\/em>2<\/em>+xy+<\/em>x<\/em>2<\/em>=2<\/em>3<\/em> <\/em>tan<\/em>-1<\/em>2y+x<\/em>3<\/em>x<\/em>+<\/em>2C1<\/em> \u00a0<\/p>\n

                                                                    log x<\/em>2<\/em>+xy+<\/em>y<\/em>2<\/em>=2<\/em>3<\/em> <\/em>tan<\/em>-1<\/em>x+2y<\/em>3<\/em>x+C<\/em> \u00a0<\/p>\n

                                                                    which is the general solution of the given differential equation.<\/p>\n

                                                                    Key Points<\/u><\/strong><\/p>\n

                                                                      \n
                                                                    • The chain rule is fundamental in calculus and used to differentiate composite functions.<\/li>\n<\/ul>\n

                                                                      \u00a0<\/p>\n

                                                                        \n
                                                                      • It provides a method to find the derivative of a function that is composed of multiple functions.<\/li>\n<\/ul>\n

                                                                        \u00a0<\/p>\n

                                                                          \n
                                                                        • The chain rule states that if y = f(g(x)), then dy\/dx = f'(g(x)) * g'(x).<\/li>\n<\/ul>\n

                                                                          \u00a0<\/p>\n

                                                                            \n
                                                                          • f'(g(x)) represents the derivative of the outer function f(u) evaluated at the inner function g(x).<\/li>\n<\/ul>\n

                                                                            \u00a0<\/p>\n

                                                                              \n
                                                                            • g'(x) represents the derivative of the inner function g(x) for x.<\/li>\n<\/ul>\n

                                                                              \u00a0<\/p>\n

                                                                                \n
                                                                              • The chain rule allows us to break down complex functions into simpler functions and differentiate them separately.<\/li>\n<\/ul>\n

                                                                                \u00a0<\/p>\n

                                                                                  \n
                                                                                • 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

                                                                                  \u00a0<\/p>\n

                                                                                    \n
                                                                                  • The chain rule can be extended to functions with multiple variables using partial derivatives.<\/li>\n<\/ul>\n

                                                                                    \u00a0<\/p>\n

                                                                                      \n
                                                                                    • Implicit differentiation uses the chain rule to find derivatives of implicitly defined functions.<\/li>\n<\/ul>\n

                                                                                      \u00a0<\/p>\n

                                                                                        \n
                                                                                      • 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>\n