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DERIVATIVES
Review- 2 Differentiation Rules
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For a function f(x) the average rate of change along the function is given by:
Which is called the derivative of f
f '(x) lim
x 0f (x x) f (x)
x
Definition of the Derivative
Notation:
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3) find the Derivative of cos x
x
xxx
xxf
coscos
0
lim)(cos'
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Trig Derivatives
x
dx
xdcos
sin
x
dx
xdsin
cos
x
dx
xd 2sectan
x
dx
xd 2csccot
xx
dx
xdtansec
sec
xx
dx
xdcotcsc
csc
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The Power Rule
If n is a rational number, then the function
is differential and
d x n dx
nxn 1
f (x) x n
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4) Find the derivative
2
25 63)(
x
xxxf
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The Sum and Difference Rule
The derivative of a sum is the sum of the derivatives
Proof pg 111
If f (x) g(x) h(x) then
f '(x) g'(x) h'(x)
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5) Find the derivative
xxxf sin5)( 3
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The Product Rule
The product of two differentiable functions f and g is itself differentiable. Moreover, the derivative of fg is the first times the derivative of the second, plus the second times the derivative of the first
Proof pg 119
(fg) '(x) = f(x) g '(x) + g(x) f '(x)
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6) Find the derivative
xxxf sin3)( 2
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The Quotient Rule
The derivative of f g is given by:The denominator times the derivative of the numerator minus the numerator times the derivative of the denominator all divided by the denominator squared
Proof pg 121
f (x) n(x)
d(x)
d(x)n'(x) n(x)d'(x)
[d(x)]2
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7) Find the derivative
53
12)(
2
x
xxf
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The Chain Rule
If g(x) is differentiable then
Power rule times the derivative of inside
Proof pg 131
d[g(x)]k
dxk[g(x)]k 1
d(g(x))
dx
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8) Find the derivative
3 22 1)( xxf
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9) Find the derivative
xxf 4sin)( 3
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HOME WORKPage 158 # 15-29 odd, 41, 45, 49, and 51, and Worksheet 2-B