Question

Differentiating a Logarithmic Function in Exercise, find the derivative of the function. See Examples 1, 2, 3, and 4.

Answer 1

Answer:

$\frac{dy}{dx}=\frac{8lnx}{x}$

Step-by-step explanation:

We are given that a function  

$y=(lnx^2)^2$

We have to find the derivative of the function  

Differentiate w.r.t x

$\frac{dy}{dx}=2(lnx^2)\times \frac{1}{x^2}\times 2x$

By using formula

$\frac{d(lnx)}{dx}=\frac{1}{x}$

$\frac{dx^n}{dx}=nx^{n-1}$

$\frac{dy}{dx}=\frac{4lnx^2}{x}=\frac{4(2)lnx}{x}$

by using $lnx^y=ylnx$

Hence, the derivative of function

$\frac{dy}{dx}==\frac{8lnx}{x}$

Answer 2

Answer: The required derivative is $\dfrac{dy}{dx}=\\dfrac{4}{x\ln x^2}$

Step-by-step explanation:

Since we have given that

$y=(\ln x^2)^2$

We need to derivative it w.r.t 'x'., using "Chain rule"

As we know that

$\dfrac{d}{dx} \ln x=\dfrac{1}{x}\\\\and\\\\\dfrac{d}{dx}x^n=nx^{n-1}$

So, it becomes,

$\dfrac{dy}{dx}=\dfrac{1}{\ln (x^2)^2}\times 2\ln(x^2)\times \dfrac{1}{x^2}\times 2x\\\\\dfrac{dy}{dx}=\dfrac{4}{x\ln x^2}$

Hence, the required derivative is $\dfrac{dy}{dx}=\\dfrac{4}{x\ln x^2}$