How to Use the Chain Rule in Calculus

In calculus, the chain rule helps problem solvers calculate and differentiate the composition of one or more functions in the same equation. Whether you are in the classroom or in a real life situation, being able to solve a complex equation using the chain rule can be more efficient than solving for a certain variable. With the knowledge of solving simple derivatives, you can implement the chain rule to solve for a product or simplify your equation.

Things You'll Need

• Calculator

Instructions

1. Separate the outer function from the inner function. For example, the equation F(x) = (x^3 + 5x)^7 would be split into the outer function: (u)^7 and the inner function (x^3 + 5x). In this example, the "u" represents the inner function.

2. Take the derivative of the inner function. For example, if the inner function is x^3 + 5x, the derivative would be 3x^2 + 5.

3. Find the derivative of the outer function. For example, if the outer function is u^7, the derivative would be 7u^6.

4. Substitute the "u" value back into the function you derived. For example, if your derived function is 7u^6, the function would now look like this: 7(x^3 + 5x)^6

5. Write the equation in chain rule format by multiplying the two derivatives together. Simplify the equation so that you can solve for the missing variable, such as "x."

References

• University of Tennessee: Visual Calculus - Chain Rule
• The Math Page: An Approach to Calculus - The Chain Rule