# 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:

$f'(u(x))=\frac{df}{du}\cdot\frac{du}{dx}=2u\cdot2x=2(x^2+5)\cdot2x=\mathbf{4x^3+20x}$

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