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olya-2409 [2.1K]
2 years ago
5

Find

frac{dy}{dx}" align="absmiddle" class="latex-formula">
\sf y=\dfrac{x}{sin^nx}

n is an integer .



Note:-

Solve with proper explanation ,

Pls don't answer if you are not sure


Spams, irrelevant, wrong answers will be deleted.​
Mathematics
2 answers:
faust18 [17]2 years ago
8 0

Hey there mate!

Please check the attached answer of picture for explanation.

Have \: a \: great \: day! \: :)

RUDIKE [14]2 years ago
6 0

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)}

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