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Section 3.4
The Chain Rule
One of THE MOST POWERFUL Rules of Differentiation
The chain rule allows you to take derivatives of compositions of functions that may be hard or even impossible to differentiate with previous rules only.
Examples: not much fun, but doable with previous
techniques
Previously impossible (Unless you want to try to definition)
42 1)( xxf
322 1)( xxf
The Chain Rule
If y=f(u) is a differentiable function of u, and
u=g(x) is a differentiable function of x, then
y=f(g(x)) is a differentiable function of x and
dy/dx = dy/du du/dx
OR
)('))((')](([ xgxgfxgfdx
d
What does this mean?
When taking the derivative of a composite
function, you first take the derivative of the
outside function at the inner function and
then multiply by the derivative of the inner
function.
Common Types
1)One of the most common types of composite functions is
What the chain rule tells us to do in this case is take
the derivative with respect to the outside exponent,
and leave the inner function alone. Then, we multiply
by the derivative of the inner function (what was being
raised to the power)
nxuy )]([
Examples:
3
32
3
8
)79(
xy
ty
What about Trig functions and Exponentials?
dx
duee
dx
ddx
duuu
dx
ddx
duuu
dx
ddx
duuu
dx
d
uu )(][
tan)(sec][sec
)(sec][tan
)(cos][sin
2
dx
duuuu
dx
ddx
duuu
dx
ddx
duuu
dx
d
)cot(csc][csc
)(csc][cot
)(sin][cos
2
Examples
xy
xxy
3tan2
sinsin
Exponentials and LogarithmsLet a be a positive real number not = 1 and
let u be a differentiable function of x.
0,1
][ln udx
du
yu
dx
d
dx
du
uu
dx
d 1][ln 0,
1][ln xx
xdx
d
dx
duaaa
dx
d
aaadx
d
uu
xx
))((ln
))((ln
dx
du
uau
dx
d
xax
dx
d
a
a
)(ln
1log
)(ln
1log
More Examples:
xy
xy
xy
3
3
3
6
sec
)2ln(
And more examples…
The chain rule can be used in conjunction with both
product and quotient rules. You will need to decide
what you should do first.
2
52)(
2
52)(
2
5
5
2
x
xxg
x
xxg
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