Theorem For Chain Rule
If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x then y = f(g(x)) is a differentiable function of x and

or equivalently,


Example
Find
for


In this case, you would divide the equation as follows:

and

The final answer is

(Larson Hostetler Edwards) Section 2.4, p.131

Guidelines for Implicit Differentiation
Differentiate both sides of the equation with respect to x.
Collect all terms involving dy/dx on the left side of the equation and move all other terms to the right side of the equation.
Factor
out of the left side of the equation.
Solve for


Example
Find

given that






(Larson Hostetler Edwards) Section 2.5, p.141

Need more examples?

Larson, Ron, Robert P. Hostetler, and Bruce H. Edwards. Calculus. 8th ed. Boston and New York: Houghton Mifflin Company, 2006.