Question

Differentiate the function

19.\(y = \ln \left| {3 - 2{x^5}} \right|\)

Short answer

The derivative of the function is \(y' = \frac{{ - 10{x^4}}}{{3 - 2{x^5}}}\).

Step-by-step solution

Derivative of logarithmic functions

The derivative of a logarithmicfunctionis shown below:

1. \(\frac{d}{{dx}}\left( {{{\log }_b}x} \right) = \frac{1}{{x\ln b}}\)
2. \(\frac{d}{{dx}}\left( {\ln x} \right) = \frac{1}{x}\)
3. \(\frac{d}{{dx}}\left( {\ln u} \right) = \frac{1}{u}\frac{{du}}{{dx}}\) or \(\frac{d}{{dx}}\left( {\ln g\left( x \right)} \right) = \frac{{g'\left( x \right)}}{{g\left( x \right)}}\)

Differentiate the function

Use the above formula with the chain rule to differentiate the function as shown below:

\(\begin{aligned}{c}y'&= \frac{d}{{dx}}\left( {\ln \left| {3 - 2{x^5}} \right|} \right)\\&= \frac{1}{{3 - 2{x^5}}} \cdot \frac{d}{{dx}}\left( {3 - 2{x^5}} \right)\\&= \frac{1}{{3 - 2{x^5}}} \cdot \left( { - 10{x^4}} \right)\\&= \frac{{ - 10{x^4}}}{{3 - 2{x^5}}}\end{aligned}\)

Thus, the derivative of the function is \(y' = \frac{{ - 10{x^4}}}{{3 - 2{x^5}}}\).