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).