Question

f (x) = -2 cosec ^ 2 (x)

Answer 1

 $\displaystyle f(x)=-2\csc^2x\\\\\frac{d}{dx}f(x)=-2\cdot2\csc^{2-1}x\cdot(-\csc x\cot x)\\\frac{d}{dx}f(x)=-4\csc x\cdot(-\csc x\cot x)\\\frac{d}{dx}f(x)=4\csc^2x\cot x\\\frac{d}{dx}f(x)=\frac{4\cos x}{\sin^2x\sin x}\\\\\frac{d}{dx}f\left(\frac16\pi\right)=\frac{4\cos\frac16\pi}{\sin^3\frac16\pi}\\\frac{d}{dx}f\left(\frac16\pi\right)=\frac{4\frac12\sqrt3}{\left(\frac12\right)^3}\\\frac{d}{dx}f\left(\frac16\pi\right)=\frac{4\sqrt3}{\left(\frac12\right)^2}\displaystyle \frac{d}{dx}f\left(\frac16\pi\right)=\frac{4\sqrt3}{\frac14}\\\boxed{\boxed{\frac{d}{dx}f\left(\frac16\pi\right)=16\sqrt3}}$