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Computing Derivatives

Computing Derivatives

Example 1

Find the derivative of the function f(x) = x2, using the limit definition of the derivative. 

Example 2

Find the derivative of the function f(x) = x3, using the limit definition of the derivative. 

Example 3

Find the derivative of the function f(x) = x4, using the limit definition of the derivative. 

Example 4

Now it's time for pattern-finding. We know the following functions and their derivatives:

f(x) = x2   f'(x) = 2x

f(x) = x3   f'(x) = 3x2

f(x) = x4   f'(x) = 4x3

What is the pattern?

Example 5

Find the derivative of the power function f(x) = x10.

Example 6

Find the derivative of the power function f(x) = x85.

Example 7

Find the derivative of the power function k(x) = x3.5.

Example 8

Find the derivative of the power function k(x) = x6.

Example 9

Find the derivative of the power function

.

Example 10

Find the derivative of the power function h(x) = x(π + e).

Example 11

Find the derivative of the power function

.

Example 12

Find the derivative of the power function 

.

Example 13

Find the derivative of the power function k(x) = x0.

Example 14

Find the derivative of the power function

.

Example 15

For the derivative f'(x) = 3x2, find a possible original function. 

Example 16

For the derivative f'(x) = 8x7, find a possible original function. 

Example 17

For the derivative f'(x) = –3x–4, find a possible original function. 

Example 18

For the derivative g'(x) = –9x–10, find a possible original function. 

Example 19

For the derivative 

,

find a possible original function. 

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