Question

Find the derivative guys pls help

Answer 1

`\qquad\longrightarrow \sf f(x) = sin^(-1) \bigg((x)/(2)\bigg)\\`

The Derivative of f(x) with respect to x-

`\qquad\longrightarrow \sf (d)/(dx) \:sin^(-1) \bigg((x)/(2)\bigg)\\`

`\qquad\longrightarrow \sf \frac{1}{\sqrt{1-\bigg((x)/(2)\bigg)^2}}\;* (d)/(dx)\:(x)/(2)\\`

`\qquad\sf\because\boxed{\sf{ \: (d)/(dx) sin^(-1)x = (1)/(√(1-x^2))}}\\`

`\qquad\longrightarrow \sf \frac{1}{\sqrt{1-\bigg((x^2)/(4)\bigg)}}\;*(1)/(2) \:x^(1-1)\\`

`\qquad \sf\because\boxed{\sf{ (d)/(dx )x^n = nx^(n-1)}}\\`

`\qquad\longrightarrow \sf \frac{1}{\sqrt{1-\bigg((x^2)/(4)\bigg)}}\;*(1)/(2) * 1\\`

`\qquad\longrightarrow \sf \frac{1}{\sqrt{1-\bigg((x^2)/(4)\bigg)}}\;*(1)/(2) \\`

`\qquad\longrightarrow \sf(1)/(2)\: \frac{1}{\sqrt{1-\bigg((x^2)/(4)\bigg)}}\\`

`\qquad\longrightarrow \underline{\sf\: \frac{1}{2\:\sqrt{1-\bigg((x^2)/(4)\bigg)}}}\\`

Henceforth, Option C) is the correct answer.