Computing Derivatives
Derivative of a Product of Functions Exercises
Example 1
Let f(x) = x + 1 and g(x) = x.
- Find f'(x).
Example 2
Let f(x) = x + 1 and g(x) = x.
- Find g'(x).
Example 3
Let f(x) = x + 1 and g(x) = x.
- Find f'(x) × g'(x).
Example 4
Let f(x) = x + 1 and g(x) = x.
- Find (f × g)'(x).
Example 5
Let f(x) = x + 1 and g(x) = x.
- Must the derivative of the function f × g be equal to the product of f' and g'?
Example 6
Find the derivative of each function. They may not all require the product rule.
- f(x) = xsin x
Example 7
Find the derivative of each function. They may not all require the product rule.
- f(x) = e^{x}cos x
Example 8
Find the derivative of each function. They may not all require the product rule.
- f(x) = xln x
Example 9
Find the derivative of each function. They may not all require the product rule.
- g(x) = 5^{x}e^{x}
Example 10
Find the derivative of each function. They may not all require the product rule.
- g(x) = (log_{2 }x)(log_{3 }x)
Example 11
Find the derivative of each function. They may not all require the product rule.
- g(x) = 5e^{x}
Example 12
Find the derivative of each function. They may not all require the product rule.
- h(x) = (x^{2} + 2x)ln x
Example 13
Find the derivative of each function. They may not all require the product rule.
- h(x) = ln x cos x
Example 14
Find the derivative of each function. They may not all require the product rule.
Example 15
Find the derivative of each function. They may not all require the product rule.
- j(x) = ln x + cos x
Example 16
Find the derivative of f(x) = (x^{2} + 2)(x^{3}-4).
- Use the product rule.
Example 17
Find the derivative of f(x) = (x^{2} + 2)(x^{3}-4).
- Rewrite f by multiplying the factors together, then take the derivative.
Example 18
- Find the derivative of
f(x) = x^{2}e^{x}sin x
thinking of the function as
f(x) = (x^{2})(e^{x}sin x).
Example 19
Find the derivative of the function
f(x) = x^{3}sin x cos x
in two different ways. Give the answer with everything multiplied out (instead of factoring out common factors).