The Chain Rule Exercises
Example 1
The function below is the composition of two other functions. Without rewriting the original function, determine the inside and outside functions.
- f(x) = cos(sin x)
Example 2
The function below is the composition of two other functions. Without rewriting the original function, determine the inside and outside functions.
- f(x) = ln (x^{3})
Example 3
The function below is the composition of two other functions. Without rewriting the original function, determine the inside and outside functions.
- f(x) = 4^{{7x}}
Example 4
The function below is the composition of two other functions. Without rewriting the original function, determine the inside and outside functions.
- f(x) = (3x^{2} + 1)^{4}
Example 5
The function below is the composition of two other functions. Without rewriting the original function, determine the inside and outside functions.
- f(x) =
Example 6
Find the derivative of the function. These do all require the chain rule.
- h(x) = (x^{5} + 4)^{{99}}
Example 7
Find the derivative of the function. These do all require the chain rule.
- h(x) = sin(ln x)
Example 8
Find the derivative of the function. These do all require the chain rule.
- h(x) = ln(ln x)
Example 9
Find the derivative of the function. These do all require the chain rule.
- h(x) = e^{{sin x}}
Example 10
Find the derivative of the function. These do all require the chain rule.
- h(x) = ln x^{3}
Example 11
Let h = (ln x)^{2}.
The following formula for h'(x) comes from applying the chain rule incorrectly. Identify the mistake(s) in the formula.
Example 12
Let h = (ln x)^{2}.
The of the following formulas for h'(x) comes from applying the chain rule incorrectly. Identify the mistake(s) in the formula.
- h'(x) = 2(ln x)
Example 13
Let h = (ln x)^{2}.
The following formula for h'(x) comes from applying the chain rule incorrectly. Identify the mistake(s) in the formula.
Example 14
Let h = (ln x)^{2}.
- Find a correct formula for h'(x).