## Derivatives: Chain Rule

For p(x) = m[n(x)] = m o n:

p'(x) = m'[n(x)]n'(x)

Example #1:

Find the derivative of (x

^{−}^{1/2})^{3}using the Chain Rule.Solution #1:

Using Chain Rule with n(x) = x

n'(x) = (−½)x

m'[n(x)] = 3(x

p'(x) = m' [n(x)]n'(x) = 3x

^{−}^{1/2}and m(x) = x^{3}it follows thatn'(x) = (−½)x

^{−}^{3/2 }andm'[n(x)] = 3(x

^{−}^{1/2})^{2}= 3x^{−}^{1}^{}p'(x) = m' [n(x)]n'(x) = 3x

^{−}^{1}(−½)x^{−3/2 }p'(x) = (−3/2)x

^{−5/2}

^{ }

Example #2:

Find the derivative of sin (1 + x

^{2}) using the Chain Rule.Solution #2:

Using Chain Rule with n(x) = 1 + x

n'(x) = 2x

and

m'[n(x)] = cos(1 + x

^{2}and m(x) = sin x it follows thatn'(x) = 2x

and

m'[n(x)] = cos(1 + x

^{2})p'(x) = m' [n(x)]n'(x) = cos(1 + x

^{2})2x

p'(x) = 2x cos(1 + x

^{2})

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