Question

cot3 log (x3)
differentiation with respect to x​

Answer 1

Given:

cot3 log (x3)

To find:

Differentiation with respect to x​

Solution:

From given, we have a function, cot³ log (x³)

differentiating the above function, we get,

d/dx [ cot³ log (x³) ]

$\dfrac{d}{dx}\left(\cot ^3\left(\log \left(x^3\right)\right)\right)$

Apply chain rule,

$\dfrac{df\left(u\right)}{dx}=\dfrac{df}{du}\cdot \dfrac{du}{dx}$

$f=u^3,\:\:u=\cot \left(\log \left(x^3\right)\right)$

$=\dfrac{d}{du}\left(u^3\right)\dfrac{d}{dx}\left(\cot \left(\log \left(x^3\right)\right)\right)$

$=3u^2\left(-\dfrac{3\csc ^2\left(\log \left(x^3\right)\right)}{x}\right)$

$\mathrm{Substitute\:back}\:u=\cot \left(\log \left(x^3\right)\right)$

$=3\cot ^2\left(\log \left(x^3\right)\right)\left(-\dfrac{3\csc ^2\left(\log \left(x^3\right)\right)}{x}\right)$

$=-\dfrac{9\csc ^2\left(\log \left(x^3\right)\right)\cot ^2\left(\log \left(x^3\right)\right)}{x}$