Question

Find
`\sf (dy)/(dx)`


`\sf y=(x)/(sin^nx)`

n is an integer .



Note:-

Solve with proper explanation ,

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Answer 1

Hey there mate!

Please check the attached answer of picture for explanation.

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

Answer 2

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

solve:

`\sf y = (d)/(dx)\left((x)/(sin^nx)\right)`

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

// apply rule:
`\sf (1)/(sinx) = csc(x)` //

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

// apply product rule:
`\sf xsinx = x * (d)/(dx) (sinx) + sin(x) *(d)/(dx) (x)` //

`\hookrightarrow \sf \bold{\sf x *(d)/(dx) (csc^n (x))+ csc^n (x) * (d)/(dx) (x)}`

Lets look into deeper differentiation separately:

`\sf we \ must \ know \ that \ (d)/(dx) (x)} = 1`

now, for
`\sf (d)/(dx) (csc^n (x))` - apply chain rule

`\rightarrow \sf n\left(\csc \left(x\right)\right)^(n-1)(d)/(dx)\left(\csc \left(x\right)\right)`

`\sf \bold \ * we \ must \ know \ that \ (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)}`