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Computations of the Derivative: The Power Rule. Gottfried Leibniz (1646-1716). Sir Isaac Newton (1642-1727). Find the Derivative of:. 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. - PowerPoint PPT Presentation
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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