Implicit Differentiation
Implicit Differentiation examples with detailed solutions are presented
Implicit Differentiation ExplainedWhen we are given a function y explicitly in terms of x, we use the rules and formulas of differentions to find the derivativedy/dx. As an example we know how to find dy/dx if y = 2 x^{3} - 2 x + 1.In some other situations, however, instead of a function given explicitly, we are given an equation including terms in y and x and we are asked to find dy/dx. For example, we are given an equation that relates y and x as follows: x y + y^{2} = 1 and asked to find dy/dx. The main idea of implicit differentiation is to differentiate both sides of the given equation and then solve the new equation obtained to find dy/dx. Implicit Differentiation ExamplesExamplesExample 1Use implicit differentiation to find the derivative dy / dx where y x + sin y = 1Solution to Example 1:
Example 2Use implicit differentiation to find the derivative dy / dx where y^{ 4} + x y^{ 2} + x = 3Solution to Example 2:
Example 3Find all points on the graph of the equationwhere the tangent lines are parallel to the line x + y = 2 Solution to Example 3:
ExercisesUse implicit differentiation to find dy/dx for each equation given below.1) x e^{ y} = 3 2) x^{ 2} + y^{ 2} = 20 3) x sin(x y) = x Solutions to the Above Exercises1) dy/dx = - 1 / x2) dy/dx = - x / y 3) dy/dx = [ 1 - sin(x y) -x y cos(x y) ] / [ x^{2} cos(x y)] More References and linksTables of Formulas for DerivativesRules of Differentiation of Functions in Calculus differentiation and derivatives |