Calculus/Chain Rule/Solutions

1. Evaluate

f'(x)

if

f(x)=(x^{2}+5)^{2}

, first by expanding and differentiating directly, and then by applying the chain rule on

f(u(x))=u^{2}

where

u=x^{2}+5

. Compare answers.

First method:

f(x)=x^{4}+10x^{2}+25

\mathbf {f'(x)=4x^{3}+20x}

Second method:

f'(u(x))={\frac {df}{du}}\cdot {\frac {du}{dx}}=2u\cdot 2x=2(x^{2}+5)\cdot 2x=\mathbf {4x^{3}+20x}

The two methods give the same answer.

2. Evaluate the derivative of

y={\sqrt {1+x^{2}}}

using the chain rule by letting

y={\sqrt {u}}

and

u=1+x^{2}

.

:

{\frac {dy}{du}}={\frac {1}{2{\sqrt {u}}}};\quad {\frac {du}{dx}}=2x

{\frac {dy}{dx}}={\frac {dy}{du}}\cdot {\frac {du}{dx}}={\frac {1}{2{\sqrt {1+x^{2}}}}}\cdot 2x=\mathbf {\frac {x}{\sqrt {1+x^{2}}}}