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* ^{3} − *xy* = 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).