Reciprocal rule

Since by the chain rule,

$$\begin{eqnarray*} \frac{\mathrm{d}y}{\mathrm{d}x} \frac{\mathrm{d}x}{\mathrm{d}y} & = & \frac{\mathrm{d}y}{\mathrm{d}y} \\ & = & 1 \end{eqnarray*}$$

We can rearrange this to find

$$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{\frac{\mathrm{d}x}{\mathrm{d}y}}$$

This result is useful when finding the derivative of a function when the derivative of its inverse function is known. For example, to find ,

$$\begin{eqnarray*} y & = & \ln{x} \\ x & = & \exp{y} \\ \frac{\mathrm{d}x}{\mat... ...{d}y}{\mathrm{d}x} & = & \frac{1}{\exp{y}} \\ & = & \frac{1}{x} \end{eqnarray*}$$