Aus Online Mathematik Brückenkurs 2

The expression is composed of two parts: first, an outer part,

\displaystyle \sin \left\{ \left. {} \right\} \right.

and then an inner part, \displaystyle \left\{ \left. {} \right\} \right.=x^{2}.

When we differentiate compound expressions, we first differentiate the outer part, \displaystyle \sin \left\{ \left. {} \right\} \right., as if \displaystyle \left\{ \left. {} \right\} \right. were the variable that we differentiate with respect to, and then we multiply with the derivative of the inner part \displaystyle \left\{ \left. {} \right\} \right.^{\prime }, so that

\displaystyle \frac{d}{dx}\sin \left\{ \left. x^{2} \right\} \right.=\cos \left\{ \left. x^{2} \right\} \right.\centerdot \left( \left\{ \left. x^{2} \right\} \right. \right)^{\prime }=\cos x^{2}\centerdot 2x