We will not prove this rule.

The rule is intuitively rather clear: if #g(x)# has a derivatie equal to #2# in #x=a# and if #f(u)# has a derivative with value #3# in #g(a)#, then a small increase #\Delta a# of #a# results in an increase of #2 \Delta a# in #g(a)# and hence, #f# applied to #u+2 \Delta a# results in an increase of about #3 \cdot 2 \Delta a=6 \Delta a#.

When working with function rules, it may be convenient to use the following notation, introduced earlier:\[\frac{\dd }{\dd x}f(g(x))=\left.\frac{\dd }{\dd x}f(x)\right|_{g(x)}\cdot \frac{\dd }{\dd x}g(x)\]Here #\left.\frac{\dd }{\dd x}f(x)\right|_{a}# stands for #f'(a)#, the derivative of #f# at the point #a#. This notation is particularly useful if #f(x)# is given by a function rule: suppose #f(x)=x^2+1# and #g(x)=\frac{3}{x}#. Then #f(g(x))= \left(\frac{3}{x}\right)^2+1# the rule reads:\[\frac{\dd}{\dd x}\left(\left(\frac{3}{x}\right)^2+1\right)=\left.\frac{\dd }{\dd x}(x^2+1)\right|_{\frac{3}{x}}\cdot \frac{\dd }{\dd x}\frac{3}{x}\tiny.\]