Derivatives The Easy Way

Constant Rule and Power Rule

We have seen the following derivatives:

1. If f(x) = c, then f '(x) = 0

2. If f(x) = x, then f '(x) = 1

3. If f(x) = x2, then f '(x) = 2x

4. If f(x) = x3, then f '(x) = 3x2

5. If f(x) = x4, then f '(x) = 4x3

This leads us the guess the following theorem.

 Theorem       d             xn  =  nxn-1      dx

Proof:

We have

Applications

Example

Find the derivatives of the following functions:

1. f(x) = 4x3 - 2x100

2. f(x) = 3x5 + 4x8 - x + 2

3.  f(x) = (x3 - 2)2

Solution

We use our new derivative rules to find

1. 12x2 - 200x99

2. 15x3+32x7-1

3. First we FOIL to get

[x6 - 4x3 + 4] '

Now use the derivative rule for powers

6x5 - 12x2

Example:

Find the equation to the tangent line to

y  =  3x3 - x + 4

at the point (1,6)

Solution:

y'  =  9x2 - 1

at x = 1 this is 8. Using the point-slope equation for the line gives

y - 6  =  8(x - 1)

or

y  =  8x - 2

Example:

Find the points where the tangent line to

y  =  x3 - 3x- 24x + 3

is horizontal.

Solution:

We find

y'  =  3x2 - 6x - 24

The tangent line will be horizontal when its slope is zero, that is, the derivative is zero.  Setting the derivative equal to zero gives:

3x2 - 6x - 24  =  0

or

x2 - 2x - 8  =  0

or

(x - 4)(x + 2)  =  0

so that

x = 4    or    x = -2

Derivative of f(x) = sin(x)

 d             sin(x)  =  cos(x)      dx

Proof:

d/dx cos(x)

 d                 cos x  =  -sin x       dx

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