Q 31. - Calculus

Question

Find the derivatives of each of functions in Exercises 17–44. In some cases it may be convenient to do some preliminary algebra.

$f\left(x\right)=\frac{\ln \left(x^5\right)}{\ln \left(x^4\right)}$

Step-by-Step Solution

Answer

The required answer is $f\text{'}\left(x\right)=0$.

Step 1. Given information.

The given function is:

$f\left(x\right)=\frac{\ln \left(x^5\right)}{\ln \left(x^4\right)}$

Step 2. Find the derivative of the given function.

$f\left(x\right)=\frac{\ln \left(x^5\right)}{\ln \left(x^4\right)}$

We know that $\log a^m=m \log a$, therefore,

$$\begin{gathered} f\left(x\right)=\frac{5 \ln x}{4 \ln x} \\ =\frac{5}{4} \end{gathered}$$

We know that the derivative of a constant term is 0, and in the given question all the terms are constant therefore the derivative is 0.

$f\text{'}\left(x\right)=0$