Q 41. - 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.

$f\left(x\right)=x^2 \ln \left(\ln x\right)$

Step-by-Step Solution

Answer

The required answer is $x^2\frac{1}{\ln x}·\frac{1}{x}+2x \ln \left(\ln x\right)$.

Step 1. Given information.

The given function is:

$f\left(x\right)=x^2 \ln \left(\ln x\right)$

Step 2. Find the derivative of the given function.

$$\begin{gathered} f\left(x\right)=x^2 \ln \left(\ln x\right) \\ f\text{'}\left(x\right)=\frac{d}{dx}\left(x^2 \ln \left(\ln x\right)\right) \\ =x^2\frac{d}{dx}\left(\ln \left(\ln x\right)\right)+\ln \left(\ln x\right)\frac{d}{dx}{x}^2 \\ =x^2\frac{1}{\ln x}\frac{d}{dx}\ln x+\ln \left(\ln x\right)·2x \\ =x^2\frac{1}{\ln x}·\frac{1}{x}+2x \ln \left(\ln x\right) \end{gathered}$$