Computations of the
Derivative: The Power Rule
Sir Isaac Newton
Gottfried Leibniz
(1642-1727)
(1646-1716)
Find the Derivative of:
f ' ( x)  2
f ( x)  2 x  2
g ( x)  3 x  3 x  3 g ' ( x )  6 x  3
2
h( x )  x  x  x  1
3
2
h' ( x)  3x  2 x  1
2
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!
x10 
7 9 1 8
x  x  5 x 7  0.33 x 6  x 5  2 x 4  42
9
2
ANS :10x9  7 x8  4x7  35x6 1.98x5  5x 4  4 2 x3
f ( x) 
ANS :
3
x
1
33 x 2
5
2
AND More Fun-ctions to
dErive!
1
1
g ( x)  3 
x 4 x3
3
3
g ' ( x)  4 
x
44 x 7
Are you ready for a challenge?
Do it any way!!!!
x 3 x  2
x
5
7
2
9
1
x 3
2
3
2
x
Rules:
Theorem 3.1
– For any constant c,
Theorem 3.2
Theorem 3.3
For any integer n > 0,
d
c0
dx
d
x 1
dx
d n
x  nx n 1
dx
DRUM ROLL PLEASE…..
Enough, STOP THE DRUM ROLL!!!!!
Theorem 3.4
For any real number r,
d r
r 1
x  rx
dx
Sum and Difference
Rules
• If f(x) and g(x) are differentiable at x
and c is any constant, then:
d
(i ) [ f ( x)  g ( x)]  f ' ( x)  g ' ( x)
dx
d
(ii ) [ f ( x)  g ( x)]  f ' ( x)  g ' ( x)
dx
d
(iii ) [cf ( x)]  cf ' ( x)
dx
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
v(t )  f ' (t )  0  20  32t  20  32t ft/s
Second compute the derivative of this function to find the
acceleration
2
a(t )  v' (t )  32 f/s
The speed in the downward direction increases 32 ft/s
every second due to gravity.
Given f(x) = x3 – 6x2 + 1
a) Find the equation of the tangent line to the curve at x = 1
y = -9x + 5
b) Find all points where the curve has a horizontal tangent
X=0 and x = 4