Implicit Differentiation

In mathematics, some equations in x and y do not explicitly define y as a function x and cannot be easily manipulated to solve for y in terms of x, even though such a function may exist. When this occurs, it is implied that there exists a function y = f( x) such that the given equation is satisfied. The technique of implicit differentiation allows you to find the derivative of y with respect to x without having to solve the given equation for y. The chain rule must be used whenever the function y is being differentiated because of our assumption that y may be expressed as a function of x.

Example 1: Find if x 2 y 3xy = 10.

Differentiating implicitly with respect to x, you find that 




Example 2: Find y′ if y = sin x + cos y.

Differentiating implicitly with respect to x, you find that


 



Example 3: Find y′ at (−1,1) if x 2 + 3 xy + y 2 = −1.

Differentiating implicitly with respect to x, you find that 




Example 4: Find the slope of the tangent line to the curve x 2 + y 2 = 25 at the point (3,−4).

Because the slope of the tangent line to a curve is the derivative, differentiate implicitly with respect to x, which yields 


 


hence, at (3,−4), y′ = −3/−4 = 3/4, and the tangent line has slope 3/4 at the point (3,−4).