Product rule

The product rule use used to differentiate functions such as -- products of functions, which can not otherwise be differentiated. Various methods to derive this rule exist. However, a neat way is provided by the aforementioned implicit differentiation technique, as follows. Put , where $u$ and $v$ are functions of $x$.

$$\begin{eqnarray*} y & = & u v \\ \ln{y} & = & \ln{u} + \ln{v} \\ \frac{1}{y} \... ...ac{\mathrm{d}u}{\mathrm{d}x} + u \frac{\mathrm{d}v}{\mathrm{d}x} \end{eqnarray*}$$

Written alternatively, . This result is known as the product rule. So to find if , we put , so . Hence, .