Aus Online Mathematik Brückenkurs 2

We can write the expression as

\displaystyle \frac{1}{x\sqrt{1-x^{2}}}=\left( x\sqrt{1-x^{2}} \right)^{-1},

and then we see that we have "something raised to \displaystyle -\text{1}", which can be differentiated one step by using the chain rule:

\displaystyle \begin{align} & \frac{d}{dx}\left( \left\{ \left. x\sqrt{1-x^{2}} \right\} \right. \right)^{-1}=-1\centerdot \left( \left\{ \left. x\sqrt{1-x^{2}} \right\} \right. \right)\centerdot \left( \left\{ \left. x\sqrt{1-x^{2}} \right\} \right. \right)^{\prime } \\ & =-\frac{1}{\left( x\sqrt{1-x^{2}} \right)^{2}}\centerdot \left( x\sqrt{1-x^{2}} \right)^{\prime } \\ & =-\frac{1}{x^{^{2}}\left( 1-x^{2} \right)}\centerdot \left( x\sqrt{1-x^{2}} \right)^{\prime } \\ \end{align}

The next step is to differentiate the product \displaystyle x\centerdot \sqrt{1-x^{2}} using the product rule:

\displaystyle \begin{align} & =-\frac{1}{x^{^{2}}\left( 1-x^{2} \right)}\centerdot \left( \left( x \right)^{\prime }\sqrt{1-x^{2}}+x\left( \sqrt{1-x^{2}} \right)^{\prime } \right) \\ & =-\frac{1}{x^{^{2}}\left( 1-x^{2} \right)}\centerdot \left( 1\sqrt{1-x^{2}}+x\left( \sqrt{1-x^{2}} \right)^{\prime } \right) \\ \end{align}

The expression \displaystyle \sqrt{1-x^{2}} is of the type "root of something", so we use the chain rule to differentiate,

\displaystyle \begin{align} & =-\frac{1}{x^{^{2}}\left( 1-x^{2} \right)}\left( \sqrt{1-x^{2}}+x\centerdot \frac{1}{2\sqrt{1-x^{2}}}\left( 1-x^{2} \right)^{\prime } \right) \\ & =-\frac{1}{x^{^{2}}\left( 1-x^{2} \right)}\left( \sqrt{1-x^{2}}+x\centerdot \frac{1}{2\sqrt{1-x^{2}}}\centerdot \left( -2x \right) \right) \\ & =-\frac{1}{x^{^{2}}\left( 1-x^{2} \right)}\left( \sqrt{1-x^{2}}-\frac{x^{2}}{\sqrt{1-x^{2}}} \right) \\ \end{align}

We write the expression on the right over a common denominator:

\displaystyle \begin{align} & =-\frac{1}{x^{^{2}}\left( 1-x^{2} \right)}\centerdot \left( \frac{\left( \sqrt{1-x^{2}} \right)^{2}-x^{2}}{\sqrt{1-x^{2}}} \right) \\ & =-\frac{1}{x^{^{2}}\left( 1-x^{2} \right)}\centerdot \left( \frac{1-x^{2}-x^{2}}{\sqrt{1-x^{2}}} \right) \\ & =-\frac{1-2x^{2}}{x^{2}\left( 1-x^{2} \right)^{{3}/{2}\;}} \\ \end{align}

NOTE: When we make simplifications of the form \displaystyle \left( \sqrt{1-x^{2}} \right)^{2}=1-x^{2}, we assume that both sides are well defined (i.e. in this case that \displaystyle x lies between \displaystyle -\text{1} and \displaystyle \text{1} ).