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Compute the derivative of (x+(1+x2)1/2)n

by RoRi

Compute the derivative of

    \[ f(x) = \left( x + \sqrt{1+x^2} \right)^n. \]

For this we want to use logarithmic differentiation. To do this first we take the log of both sides,

    \begin{align*} f(x) = \left( x + \sqrt{1+x^2} \right)^n && \implies && \log (f(x)) &= \log \left(\left( x + \sqrt{1+x^2} \right)^n\right) \\ && \implies && \log (f(x)) &= n \log \left(x + \sqrt{1+x^2} \right). \end{align*}

Now to use logarithmic differentiation we take the derivative of both sides,

    \begin{align*} \frac{f'(x)}{f(x)} &= n \cdot \frac{1}{x+\sqrt{1+x^2}} \cdot \left(1 + \frac{x}{\sqrt{1+x^2}} \right) \\ &= n \cdot \frac{1}{x+\sqrt{1+x^2}} \cdot \left( \frac{\sqrt{1+x^2} + x}{\sqrt{1+x^2}} \right) \\ &= n \cdot \frac{1}{\sqrt{1+x^2}}. \end{align*}

Therefore, bringing f(x) over to the right to solve for f'(x) we have

    \begin{align*} f'(x) &= f(x) \cdot \frac{n}{\sqrt{1+x^2}} \\ &= \frac{n \left( x + \sqrt{1+x^2} \right)^n}{\sqrt{1+x^2}}. \end{align*}

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