Aus Online Mathematik Brückenkurs 2

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

\displaystyle \sin \bbox[#FFEEAA;,1.5pt]{\,\phantom{x^2_2}\,}

and then an inner part, \displaystyle \bbox[#FFEEAA;,1.5pt]{\,\phantom{x^2_2}\,} = x^{2}\,.

When we differentiate compound expressions, we first differentiate the outer part, \displaystyle \sin \bbox[#FFEEAA;,1.5pt]{\,\phantom{xx}\,}, as if \displaystyle \bbox[#FFEEAA;,1.5pt]{\,\phantom{xx}\,} were the variable that we differentiate with respect to, and then we multiply with the derivative of the inner part \displaystyle \bigl(\bbox[#FFEEAA;,1.5pt]{\,\phantom{xx}\,}\bigr)', so that

\displaystyle \frac{d}{dx}\,\sin \bbox[#FFEEAA;,1.5pt]{\,x^2\,} = \cos \bbox[#FFEEAA;,1.5pt]{\,x^2\,}\cdot \bigl(\bbox[#FFEEAA;,1.5pt]{\,x^2\,}\bigr)' = \cos x^2\cdot 2x\,\textrm{.}