# Computing Derivatives

### Example 1

Find the derivative of the function *f*(*x*) = *x*^{2}, using the limit definition of the derivative.

### Example 2

Find the derivative of the function *f*(*x*) = *x*^{3}, using the limit definition of the derivative.

### Example 3

Find the derivative of the function *f*(*x*) = *x*^{4}, 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*) = *x*^{2 }*f'*(*x*) = 2*x*

*f*(*x*) = *x*^{3 }*f'*(*x*) = 3*x*^{2}

*f*(*x*) = *x*^{4 }*f'*(*x*) = 4*x*^{3}

What is the pattern?

### Example 5

Find the derivative of the power function *f*(*x*) = *x*^{10}.

### Example 6

Find the derivative of the power function *f*(*x*) = *x*^{85}.

### Example 7

Find the derivative of the power function *k*(*x*) = *x*^{3.5}.

### Example 8

Find the derivative of the power function *k*(*x*) = *x ^{–}*

^{6}.

### 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*) = *x*^{0}.

### Example 14

Find the derivative of the power function

.

### Example 15

For the derivative *f'*(*x*) = 3*x*^{2}, find a possible original function.

### Example 16

For the derivative *f'*(*x*) = 8x^{7}, find a possible original function.

### Example 17

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

### Example 18

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

### Example 19

For the derivative

,

find a possible original function.