Question

Differentiate the followings with respect to x
$( {sin}^{2}(x)) \times lnx$
​

Answer 1

Answer:

$\sf{\dfrac{ { \sin }^{2} x}{x} + ln(x) \sin(2x) }$

Explanation:

Given,

$y = { \sin }^{2} x \times ln(x)$

To find the dy/dx.

We know that,

$\frac{d}{dx} mn = m \frac{dn}{dx} + n \frac{dm}{dx}$

Therefore, we get,

$= > \frac{dy}{dx} = { \sin }^{2} x \frac{d}{dx} ln(x) + ln(x) \frac{d}{dx} { \sin}^{2} x$

But, we know that,

• $\frac{d}{dx} ln(x) = \frac{1}{x}$

• $\frac{d}{dx} { \sin }^{2} x = 2 \sin(x) \cos(x) = \sin (2x)$

Therefore, we get,

$= > \frac{dy}{dx} = { \sin}^{2} x \times \frac{1}{x} + ln(x) \sin(2x) \\ \\ = > \frac{dy}{dx} = \frac{ { \sin }^{2} x}{x} + ln(x) \sin(2x)$

Hence, the required value is $\bold{ \dfrac{ { \sin }^{2} x}{x} + ln(x) \sin(2x) }$