91Ó°ÊÓ

Recall that the derivative of \(\cot x\) is \(- \csc^2 x\). Thus, the first derivative of \(y = \cot x\) is: $$y^{\prime} = -\csc^2 x$$

To find the second derivative, we will differentiate \(y^{\prime}\) with respect to \(x\). Using the formula for the derivative of \(\csc u\), we have: $$\frac{d}{dx}(\csc u) = -\csc u \cot u \frac{du}{dx}$$ So, we differentiate \(-\csc^2 x\) with respect to \(x\): First, let's find the derivative of \(\csc^2 x\). Setting \(u = \csc x\), then we have: $$\frac{d}{dx}(\csc^2 x) = \frac{d}{dx}(u^2) = 2u \frac{du}{dx}$$ Substitute \(\csc x\) for \(u\) and substitute \(\frac{d}{dx}(\csc x)\) according to the previously mentioned formula: $$\frac{d}{dx}(\csc^2 x) = 2(\csc x) \left(-\csc x \cot x \right) = -2 \csc^2 x \cot x $$ Now we can find the second derivative \(y^{\prime \prime}\) by finding the derivative of \(-\csc^2 x\): $$y^{\prime \prime} = \frac{d}{dx}(-\csc^2 x) = -(-2 \csc^2 x \cot x) = 2\csc^2 x \cot x$$ The second derivative of the given function is: $$y^{\prime \prime} = 2\csc^2 x \cot x$$