Download 3_04_ChainRule

Math 1304 Calculus I
3.4 – The Chain Rule
Ways of Stating The Chain Rule
• Statements of chain rule:
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If F = fog is a composite, defined by F(x) =
f(g(x)) then
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F'(x) = f'(g(x))g'(x)
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If y = f(u) and u = g(x) are differentiable, then
dy/du = dy/dx dx/du
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Other ways of writing it
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(fog)'(x) = f'(g(x)) g'(x)
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Basic ideas - for a chain of functions, rates
multiply together
The Chain Rule
• The derivative of the composition is…
f
g
The Chain Rule
• The derivative of the composition is the
product of the derivatives
f’
g’
f
g
z
y
x
fg
(f  g)’=f’ g’
The Chain Rule
• The derivative of the composition is the
product of the derivatives
f’
g’
f
g
z
y
x
fg
(f  g)’=f’ g’
Proof of Chain Rule
• Use the notation dy/dx, show that if y=g(x)
and z=f(y), then dz/dx = dz/dy dy/dx
Proof :
 z y 
dz
z
z
y
z
y dz dy
  lim
 lim
 lim 
lim
 lim
lim


x

0

x

0

x

0

x

0

y

0

x

0
dx
x
y
x
y
x dy dx
 y x 
Basic Approach to Chain Rule
• Identify inside and outside functions
• Take the derivative of outside function (put
inside function inside, as is)
• Multiply by derivative of inside function
A good working set of rules
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Constants: If f(x) = c, then f’(x) = 0
Powers:
If f(x) = xn, then f’(x) = nxn-1
Exponentials: If f(x) = ax, then f’(x) = (ln a) ax
Trigonometric functions:
If f(x) = sin(x), then f’(x) = cos(x)
If f(x) = cos(x), then f’(x) = - sin(x)
Scalar mult: If f(x) = c g(x), then f’(x) = c g’(x)
Sum:
If
f(x) = g(x) + h(x),
then f’(x) = g’(x) + h’(x)
Difference: If
f(x) = g(x) - h(x),
then f’(x) = g’(x) - h’(x)
Multiple sums: derivative of sum is sum of derivatives
Linear combinations: derivative of linear combo is linear combo of derivatives
Product:
If
f(x) = g(x) h(x),
then f’(x) = g’(x) h(x) + g(x)h’(x)
Multiple products: If
f(x) = g(x) h(x) k(x),
then f’(x) = g’(x) h(x) k(x) + g(x) h’(x) k(x) + g(x) h(x) k’(x)
Quotient:
If
f(x) = g(x)/h(x),
then f’(x) = (g’(x) h(x) - g(x)h’(x))/(h(x))2
Composition: If F = fog is a composite, defined by F(x) = f(g(x))
then F'(x) = f'(g(x))g'(x)
A new good working set of rules
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Constants: If F(x) = c, then f’(x) = 0
Powers:
If F(x) = f(x)n, then F’(x) = n f(x)n-1 f’(x)
Exponentials: If F(x) = af(x), then F’(x) = (ln a) af(x) f’(x)
All trigonometric functions:
If F(x) = sin(f(x)), then F’(x) = cos(f(x)) f’(x)
If F(x) = cos(f(x)), then F’(x) = - sin(f(x)) f’(x)
Scalar mult: If F(x) = c f(x), then F’(x) = c f’(x)
Sum:
If
F(x) = g(x) + h(x),
then F’(x) = g’(x) + h’(x)
Difference: If
F(x) = g(x) - h(x),
then F’(x) = g’(x) - h’(x)
Multiple sums: derivative of sum is sum of derivatives
Linear combinations: derivative of linear combo is linear combo of derivatives
Product:
If
F(x) = g(x) h(x),
then F’(x) = g’(x) h(x) + g(x)h’(x)
Multiple products: If
F(x) = g(x) h(x) k(x),
then F’(x) = g’(x) h(x) k(x) + g(x) h’(x) k(x) + g(x) h(x) k’(x)
Quotient:
If
F(x) = g(x)/h(x),
then F’(x) = (g’(x) h(x) - g(x)h’(x))/(h(x))2
Composition: If F = fog is a composite, defined by F(x) = f(g(x))
then F'(x) = f'(g(x))g'(x)