Question

Differentiate the function.

25.\(y = \ln \frac{{{x^a}}}{{{b^x}}}\)

Short answer

The derivative of the function is \(y' = \frac{a}{x} - \ln b\).

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

Rewrite the function as shown below:

Use the formula to differentiate the function as shown below:

\(\begin{aligned}{c}y'&= \frac{d}{{dx}}\left( {a\ln x - x\ln b} \right)\\&= a \cdot \frac{1}{x} - \ln b \cdot 1\\&= \frac{a}{x} - \ln b\end{aligned}\)

Thus, the derivative of the function is \(y' = \frac{a}{x} - \ln b\).