next up previous
Next: Quotient rule Up: Rules of differentiation Previous: Implicit differentiation

Product rule

The product rule use used to differentiate functions such as \( y =
x \exp{x} \) -- 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 \( y = u v \), 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, \( (uv)' = u'v + uv' \). This result is known as the product rule. So to find \(
\frac{\mathrm{d}y}{\mathrm{d}x} \) if \( y =
x \exp{x} \), we put \( v = x, u = \exp{x} \), so \(
\frac{\mathrm{d}v}{\mathrm{d}x} = 1, \frac{\mathrm{d}u}{\mathrm{d}x} = \exp{x} \). Hence, \( \frac{\mathrm{d}y}{\mathrm{d}x}
= x \exp{x} + \exp{x} = \exp{x} (x + 1) \).

Alexander Frolkin 2001-03-13