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# The Chain Rule for AP Calculus

based on 2 ratings
By McGraw-Hill Professional
Updated on Oct 24, 2011

Practice problems for these concepts can be found at: Differentiation Practice Problems for AP Calculus

If y = f (u) and u = g (x ) are differentiable functions of u and x respectively, then .

### Example 1

If y = (3x – 5)10, find .

Using the chain rule, let u = 3 x – 5 and thus, y = u10. Then, and . Since . Or you can use your calculator and enter d((3x – 5)10, x ) and obtain the same result.

### Example 2

If , find f (x). Rewrite Using the product rule, .

To find , use the chain rule and let u = 25 – x2.

Thus, . Substituting this quantity back into f '(x), you have Or you can use your calculator and enter and obtain the same result.

### Example 3

If , find .

Using the chain rule, let , Then .

To find , use the quotient rule.

Thus, . Substituting this quantity back into .

An alternate solution is to use the product rule and rewrite y = as and use the quotient rule. Another approach is to express y = (2x – 1)3(x–6) and use the product rule. Of course, you can always use your calculator if you are permitted to do so.

Practice problems for these concepts can be found at: Differentiation Practice Problems for AP Calculus

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