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 $u=1+x^{2}$.

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

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