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Find multiple derivatives of a given function

by RoRi

Let

    \[ f(x) = \frac{1 - \sqrt{x}}{1+\sqrt{x}}. \]

Find formulas for the first, second, and third derivatives of f.

These are straightforward computations using the rules for derivatives of quotients and for rational powers of x.

The first derivative:

    \begin{align*} Df(x) &= \frac{\left(1+\sqrt{x}\right)\left( \frac{-1}{2 \sqrt{x}} \right) - \left( 1 - \sqrt{x} \right)\left(\frac{1}{2 \sqrt{x}} \right)}{\left( 1 + \sqrt{x} \right)^2} \\ &= \frac{-\frac{1}{2\sqrt{x}} - \frac{1}{2} - \frac{1}{2\sqrt{x}} + \frac{1}{2}}{\left( 1 + \sqrt{x} \right)^2} \\ &= \frac{-1}{\sqrt{x} \left( 1 + \sqrt{x} \right)^2}. \end{align*}

The second derivative:

    \begin{align*} D^2 f(x) &= D \left( \frac{-1}{\sqrt{x} \left( 1 + \sqrt{x} \right)^2} \right) \\ &= \frac{\frac{1}{2\sqrt{x}} \left( 1 + \sqrt{x} \right)^2 + 2 \sqrt{x} \left( 1 + \sqrt{x} \right) \left( \frac{1}{2 \sqrt{x}} \right)}{x \left( 1 + \sqrt{x} \right)^4} \\ &= \frac{1 + 3 \sqrt{x}}{2 \left(x + \sqrt{x} \right)^3}. \end{align*}

And, the third derivative:

    \begin{align*} D^3 f(x) &= D \left( \frac{1 + 3 \sqrt{x}}{2 \left( x + \sqrt{x} \right)^3} \right) \\ &= \frac{\frac{3}{2 \sqrt{x}} \left( 2 \left(x + \sqrt{x}\right)^3\right) - \left(1 + 3 \sqrt{x} \right) \left(6 \left( x + \sqrt{x} \right)^2 \left( 1 + \frac{1}{2 \sqrt{x}} \right) \right)}{4 \left( x + \sqrt{x} \right)^6} \\ &= \left( -\frac{3}{4} \right) \left( \frac{1 + 4 \sqrt{x} + 5x}{\sqrt{x} \left(x + \sqrt{x} \right)^4} \right). \end{align*}

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