Question

Find the derivative of​

Answer 1

See the solution given in the image.

Answer 2

`-(3\csc^2(x)\cos\left(√(\cot^3(x))\right)√(\cot(x)))/(2)`

Explanation:

Given:

`y=\sin\left(√(\cot^3(x))\right)`

To find the derivative, we can use the chain rule.

`\boxed{\begin{array}{c}\underline{\text{Chain Rule for Differentiation}}\\\\\text{If\;\;$y=f(u)$\;\;and\;\;$u=g(x)$\;\;then:}\\\\\frac{\text{d}y}{\text{d}x}=\frac{\text{d}y}{\text{d}u}*\frac{\text{d}u}{\text{d}x}\end{array}}`

`\textsf{Let}\;\;y=\sin(u) \;\;\textsf{where}\;\;u=√(\cot^3(x))`

Differentiate the two parts separately:

`y=\sin(u) \implies \frac{\text{d}y}{\text{d}u}=\cos(u)`

`\begin{aligned}u=√(\cot^3(x))=\left(\cot (x)\right)^(3)/(2)\implies \frac{\text{d}u}{\text{d}x}&=(3)/(2)\left(\cot (x)\right)^{(1)/(2)}\left(-\csc^2(x)\right)\\\\\frac{\text{d}u}{\text{d}x}&=-(3\csc^2(x)√(\cot (x)))/(2)\end{aligned}`

Now, put everything back into the chain rule formula:

`\frac{\text{d}y}{\text{d}x}=\cos(u)* -(3\csc^2(x)√(\cot(x)))/(2)`

`\frac{\text{d}y}{\text{d}x}=\cos\left(√(\cot^3(x))\right)* -(3\csc^2(x)√(\cot(x)))/(2)`

`\frac{\text{d}y}{\text{d}x}=-(3\csc^2(x)\cos\left(√(\cot^3(x))\right)√(\cot(x)))/(2)`