Download - 2.4 The Chain Rule
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2.4 The Chain Rule
"I recoil with dismay and horror at this lamentable plague of functions which do not have derivatives. -- Charles Hermite
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Objectives:
To find derivatives using the chain rule
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Two ways
Find the derivative of 2)56( xy
Power Product
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A third way…
Chain rule
y’ = deriv of outside (keep inside same) times deriv of inside
Example:
)('))((''))((
xgxgfyxgfy
2)56( xy
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A fourth way
Another form of the chain rule:
Example
')('
)()())((
uufdxdy
xguufyxgfy
2)56( xy
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Example
2)12( xy
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Another example
32 )1( xy
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What about trig functions?
Think of u substitution
xxf 2sin)(
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You try:32 )23( xxy 5)(sin xy
)sin( 5xy 4)(Bobdxd
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More challenging examples…
3 22 )1( xy
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Another one
2)32(7)(
xxg What rules will we use?
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Yet one more…22 1 xxy
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Review:
Chain Rule:
)()('
)(1 quantityofderivquantitykny
quantitykyn
n
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2 more examples
3 2 4
x
xy
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Last one2
2 313
xxy
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Trig Chain Rule
Deriv of trig (keep angle same) deriv of angle
Be careful of double chain rule…
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Example
xy 2sin41 2
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Another example
ty 4sin3
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Three types…
)3cos( 2xy 2)3cos( xy
2))3(cos( xy