Differentiation Rules

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/dx (xn) = nxn-1

Proof:

First note that the binomial theorem says that

(x + h)n  =  xn + nhxn-1 + nC2h2xn-2 + ... + nhn-1x + hn

=   xn + nhxn-1 + h2(Stuff)

For our purposes, stuff is some polynomial in h and k that we do not care about.

We have

Sum Difference and Constant Multiple Rules

Theorem:  If f and g are differentiable then

A)   [f + g]' = f ' + g'

B)  [f - g]' = f ' - g'

C)  [cf]' = c(f ')

Proof of C)

cf(x + h) - cf(x)
lim
h ->0              h

c[f(x + h) - f(x)]
=  lim
h ->0              h

f(x + h) - f(x)
=  c lim
h ->0              h

=  c f '(x)

Example

We know that if f(x) = x2 then f '(x) = 2x and that if f(x) = x then f '(x) = 1 so that if

h(x) = 4x2 -3x

then

h'(x)  =  4(2x) - 3(1)  =  8x - 1

More 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

4. f(x)  =  10

5. f(x)  =  6 / x3

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

4. Begin by writing the square root sign as a fractional exponent

10x1/2

Now use the power rule to get

f '(x)  =  10(1/2)x -1/2  =  5x -1/2

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

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