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### Section 3.13 : Logarithmic Differentiation

4. Find the first derivative of $$g\left( w \right) = {\left( {3w - 7} \right)^{4w}}$$.

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We just need to do some logarithmic differentiation so take the logarithm of both sides and do a little simplifying.

$\ln \left[ {g\left( w \right)} \right] = \ln \left[ {{{\left( {3w - 7} \right)}^{4w}}} \right] = 4w\,\,\ln \left( {3w - 7} \right)$ Show Step 2

Use implicit differentiation to differentiate both sides with respect to $$w$$. Don’t forget to product rule the right side.

$\frac{{g'\left( w \right)}}{{g\left( w \right)}} = 4\ln \left( {3w - 7} \right) + 4w\frac{3}{{3w - 7}} = 4\ln \left( {3w - 7} \right) + \frac{{12w}}{{3w - 7}}$ Show Step 3

Finally, solve for the derivative and plug in the equation for $$g\left( w \right)$$ .

\begin{align*}g'\left( w \right) & = g\left( w \right)\left[ {4\ln \left( {3w - 7} \right) + \frac{{12w}}{{3w - 7}}} \right]\\ & = \require{bbox} \bbox[2pt,border:1px solid black]{{{{\left( {3w - 7} \right)}^{4w}}\left[ {4\ln \left( {3w - 7} \right) + \frac{{12w}}{{3w - 7}}} \right]}}\end{align*}