Question

Find $\sf \dfrac{dy}{dx}$

$\sf y=\dfrac{x}{sin^nx}$

n is an integer .



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Answer 1

Hey there mate!

Please check the attached answer of picture for explanation.

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Answer 2

Answer:

$\sf \dfrac{dy}{dx} =\sf \bold{ -nx\cot \left(x\right)\csc ^n\left(x\right) + \csc ^n\left(x\right)}$

solve:

$\sf y = \dfrac{d}{dx}\left(\dfrac{x}{sin^nx}\right)$

$\hookrightarrow \sf \bold{ \sf \dfrac{dy}{dx} =\sf \dfrac{d}{dx}\left(\dfrac{x}{sin^nx}\right)}$

// apply rule:  $\sf \dfrac{1}{sinx} = csc(x)$ //

$\hookrightarrow \sf \bold{ \sf \dfrac{dy}{dx} =\sf \dfrac{d}{dx}\left(x\csc ^n\left(x\right)\right)}$

// apply product rule:  $\sf xsinx = x * \frac{d}{dx} (sinx) + sin(x) *\frac{d}{dx} (x)$ //

$\hookrightarrow \sf \bold{\sf x *\frac{d}{dx} (csc^n (x))+ csc^n (x) * \dfrac{d}{dx} (x)}$

Lets look into deeper differentiation separately:

$\sf we \ must \ know \ that \ \dfrac{d}{dx} (x)} = 1$

now, for $\sf \frac{d}{dx} (csc^n (x))$  - apply chain rule

$\rightarrow \sf n\left(\csc \left(x\right)\right)^{n-1}\dfrac{d}{dx}\left(\csc \left(x\right)\right)$

 $\sf \bold \ * we \ must \ know \ that \ \dfrac{d}{dx} (csc(x)) = -cot(x) csc(x)$

$\sf \rightarrow n\left(\csc \left(x\right)\right)^{n-1}\left(-\cot \left(x\right)\csc \left(x\right)\right)$

$\rightarrow \sf -n\cot \left(x\right)\csc ^{n-1+1}\left(x\right)$

$\rightarrow \sf -n\cot \left(x\right)\csc ^n\left(x\right)$

Now finish:

$\hookrightarrow \sf x * -n\cot \left(x\right)\csc ^n\left(x\right) + \csc ^n\left(x\right) *1$

$\hookrightarrow \sf \bold{ -nx\cot \left(x\right)\csc ^n\left(x\right) + \csc ^n\left(x\right)}$