# logarithmic proof of product rule

###### Proof.

Let $f$ and $g$ be differentiable functions and $y=f(x)g(x)$. Then $\ln y=\ln(f(x)g(x))=\ln f(x)+\ln g(x)$. Thus, $\displaystyle\frac{1}{y}\cdot\frac{dy}{dx}=\frac{f^{\prime}(x)}{f(x)}+\frac{g^% {\prime}(x)}{g(x)}$. Therefore,

$\begin{array}[]{rl}\displaystyle\frac{dy}{dx}&\displaystyle=y\left(\frac{f^{% \prime}(x)}{f(x)}+\frac{g^{\prime}(x)}{g(x)}\right)\\ &\\ &\displaystyle=f(x)g(x)\left(\frac{f^{\prime}(x)}{f(x)}+\frac{g^{\prime}(x)}{g% (x)}\right)\\ &\\ &=f^{\prime}(x)g(x)+g^{\prime}(x)f(x).\end{array}$

Once students are familiar with the natural logarithm, the chain rule, and implicit differentiation, they typically have no problem following this proof of the product rule. Actually, with some prompting, they can produce a proof of the product rule to this one. This exercise is a great way for students to review many concepts from calculus  .

Title logarithmic proof of product rule LogarithmicProofOfProductRule 2013-03-22 16:18:48 2013-03-22 16:18:48 Wkbj79 (1863) Wkbj79 (1863) 6 Wkbj79 (1863) Proof msc 26A06 msc 97D40 LogarithmicProofOfQuotientRule