Calculus/Chain Rule/Solutions

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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}}