Computations of the Derivative: The Power Rule

Computations of the Computations of the Derivative: The Power RuleDerivative: The Power Rule

Computations of the Computations of the Derivative: The Power RuleDerivative: The Power Rule

Sir Isaac Newton

(1642-1727)

Gottfried Leibniz

(1646-1716)

Find the Derivative of: 22)( xxf

333)( 2 xxxg

1)( 23 xxxxh

123)(' 2 xxxh

2)(' xf

36)(' xxg

What’s going on????

• Write your own rule! On your sheet of paper come up with a rule that can be used to derive polynomials.

Now try doing a lot with a Little!

42233.052

1

9

7 45678910 xxxxxxx

3456789 24598.1354710: xxxxxxxANS

53 2)( xxf

3 23

1:

xANS

AND More Fun-ctions to dErive!

4 33

11)(

xxxg

4 744

33)('

xxxg

x

xx 235

3 2

2

7 13

2

9

xx

Are you ready for a challenge?

Do it any way!!!!

Rules:Theorem 3.1

– For any constant c,

0cdx

d

1xdx

dTheorem 3.2

Theorem 3.3

For any integer n > 0,

1 nn nxxdx

d

DRUM ROLL PLEASE…..

1 rr rxxdx

d

Theorem 3.4For any real number r,

Enough, STOP THE DRUM ROLL!!!!!

Sum and Difference Rules

• If f(x) and g(x) are differentiable at x and c is any constant, then:

)(')]([)(

)(')(')]()([)(

)(')(')]()([)(

xcfxcfdx

diii

xgxfxgxfdx

dii

xgxfxgxfdx

di

Examples:• Suppose that the height of a skydiver t seconds after

jumping from an airplane is given by f(t) = 225 – 20t – 16t2 feet. Find the person’s acceleration at time t.

First compute the derivative of this function to find the velocity

Second compute the derivative of this function to find the acceleration

The speed in the downward direction increases 32 ft/s every second due to gravity.

ft/s 322032200)(')( tttftv

2f/s 32)(')( tvta

Given f(x) = x3 – 6x2 + 1

a) Find the equation of the tangent line to the curve at x = 1

b) Find all points where the curve has a horizontal tangent

y = -9x + 5

X=0 and x = 4