Derivative

• The derivative of $f(x)$ at $x=a$, aka $f^{\prime }(a)$, is $$
  f'(a)=\lim_{ h \to 0 } \left( \frac{f(a+h)-f(a)}{h} \right)=\lim_{ x \to a } \left( \frac{f(x)-f(a)}{x-a} \right)

\begin{align}
x & =f(y) \
1 & =f'(y)y' \
y' & =\frac{1}{f'(y)} \
& =\frac{1}{f'(f^{-1}(x))}
\end

\begin{align}
V & =\frac{4}{3}\pi r^{3} \
\frac{ \mathrm{d}V }{ \mathrm{d}r } & =4\pi r^{2} \
\mathrm{d}V & =4\pi r^{2}\mathrm{d}r \
\frac{\mathrm{dV}}{V} & =\frac{4\pi r^{2}\mathrm{d}r}{\frac{4}{3}\pi r^{3}} \
& =\frac{3}{r}\mathrm{d}r \
& =\frac{3}{21}(0.07) \
& =1%
\end