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Section 3.13 : Logarithmic Differentiation

5. Find the first derivative of \(f\left( x \right) = {\left( {2x - {{\bf{e}}^{8x}}} \right)^{\sin \left( {2x} \right)}}\).

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

\[\ln \left[ {f\left( x \right)} \right] = \ln \left[ {{{\left( {2x - {{\bf{e}}^{8x}}} \right)}^{\sin \left( {2x} \right)}}} \right] = \sin \left( {2x} \right)\ln \left( {2x - {{\bf{e}}^{8x}}} \right)\] Show Step 2

Use implicit differentiation to differentiate both sides with respect to \(x\). Don’t forget to product rule the right side.

\[\begin{align*}\frac{{f'\left( x \right)}}{{f\left( x \right)}} & = 2\cos \left( {2x} \right)\ln \left( {2x - {{\bf{e}}^{8x}}} \right) + \sin \left( {2x} \right)\frac{{2 - 8{{\bf{e}}^{8x}}}}{{2x - {{\bf{e}}^{8x}}}}\\ & = 2\cos \left( {2x} \right)\ln \left( {2x - {{\bf{e}}^{8x}}} \right) + \sin \left( {2x} \right)\frac{{2 - 8{{\bf{e}}^{8x}}}}{{2x - {{\bf{e}}^{8x}}}}\end{align*}\] Show Step 3

Finally, solve for the derivative and plug in the equation for \(f\left( x \right)\) .

\[\begin{align*}f'\left( x \right) & = f\left( x \right)\left[ {2\cos \left( {2x} \right)\ln \left( {2x - {{\bf{e}}^{8x}}} \right) + \sin \left( {2x} \right)\frac{{2 - 8{{\bf{e}}^{8x}}}}{{2x - {{\bf{e}}^{8x}}}}} \right]\\ & = \require{bbox} \bbox[2pt,border:1px solid black]{{{{\left( {2x - {{\bf{e}}^{8x}}} \right)}^{\sin \left( {2x} \right)}}\left[ {2\cos \left( {2x} \right)\ln \left( {2x - {{\bf{e}}^{8x}}} \right) + \sin \left( {2x} \right)\frac{{2 - 8{{\bf{e}}^{8x}}}}{{2x - {{\bf{e}}^{8x}}}}} \right]}}\end{align*}\]