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Computing Derivatives

Computing Derivatives

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Introduction to Computing Derivatives - At A Glance:

Use the product rule whenever there is a function that's a product of two other functions of x. If

f(x) = ({something with an x}) × ({something else with an x}),

then use the product rule to find f'.

To use the product rule, figure out which two functions are multiplied together, and the derivative of each of those functions.

Example 1

Find the derivative of h(x) = x2ex.


Example 2

Find the derivative of f(x) = sin x cos x.


Example 3

Find the derivative of g(x) = x2sin x


Example 4

Find the derivative of f(x) = 4x2.


Example 5

Find the derivative of

f(x) = x2exsin x.


Exercise 1

Let f(x) = x + 1 and g(x) = x

  • Find f'(x).

Exercise 2

Let f(x) = x + 1 and g(x) = x.

  • Find g'(x).

Exercise 3

Let f(x) = x + 1 and g(x) = x.

  • Find f'(x) × g'(x).

Exercise 4

Let f(x) = x + 1 and g(x) = x.

  • Find (f × g)'(x).

Exercise 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'?

Exercise 6

Find the derivative of each function. They may not all require the product rule.

  • f(x) = xsin x

Exercise 7

Find the derivative of each function. They may not all require the product rule.

  • f(x) = excos x

Exercise 8

Find the derivative of each function. They may not all require the product rule.

  • f(x) = xln x

Exercise 9

Find the derivative of each function. They may not all require the product rule.

  • g(x) = 5xex

Exercise 10

Find the derivative of each function. They may not all require the product rule.

  •  g(x) = (logx)(logx)

Exercise 11

Find the derivative of each function. They may not all require the product rule.

  • g(x) = 5ex

Exercise 12

Find the derivative of each function. They may not all require the product rule.

  • h(x) = (x2 + 2x)ln x

Exercise 13

Find the derivative of each function. They may not all require the product rule.

  • h(x) = ln cos x

Exercise 14

Find the derivative of each function. They may not all require the product rule.

Exercise 15

Find the derivative of each function. They may not all require the product rule.

  • j(x) = ln x + cos x

Exercise 16

Find the derivative of f(x) = (x2 + 2)(x3-4).

  • Use the product rule.

Exercise 17

Find the derivative of f(x) = (x2 + 2)(x3-4).

  • Rewrite f by multiplying the factors together, then take the derivative.

Exercise 18

  • Find the derivative of

f(x) = x2exsin x

thinking of the function as

f(x) = (x2)(exsin x).

Exercise 19

Find the derivative of the function

f(x) = x3sin cos x

in two different ways. Give the answer with everything multiplied out (instead of factoring out common factors).

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