Question

find dy/dx if y= log3(log5^x)​

Answer 1

Answer:

Step-by-step explanation:

log=15*15

log=125

Answer 2

Concept:

To answer this question we need to recall the concept of differential in calculus.

• Derivative of a quantity refers to the rate change per unit time.

• In mathematics, differential represents the principle part of change in a function y = f(x) with respect to changes in x, where x is independent variable and y is dependent variable.

Given:

The function y = $log_{3}(log_{5}x)$

To find:

The differential of y with respect to x i.e. dy/dx.

Solution:

By using the formula of log :

 $log_{a}b=\frac{log_{e} b}{log_{e}a}$     and  $log(\frac{b}{a})= log b -log a$

        y = $log_{3}(\frac{logx}{log 5})$

        y = $log_{3}({logx})-log_{3}({log 5})$

        y = $\frac{log({logx})}{log 3}-log_{3}({log 5})$

Now differentiating y with respect to x.

      $\frac{dy}{dx} =\frac{d}{dx}$ $(\frac{log({logx})}{log 3}-log_{3}({log 5}))$

           =  $\frac{1}{log 3 } \frac{d}{dx}[log(log x)]- \frac{d}{dx}[log_{3}(log 5)]$

           =  $\frac{1}{log 3 } \frac{1}{log x }[\frac{d}{dx}(log x)]- 0$               (since,   $[log_{3}(log 5)]$ is constant term)

           =  $\frac{1}{log 3}$  ×  $\frac{1}{log x}$  ×  $\frac{1}{x}$

Hence, $\frac{dy}{dx} =$ $\frac{1}{xlogxlog3}$ is the required answer.