Find $y^{\prime \prime }$ if $y=4\sec x$

Step 1 ：We are given the function $y=4\sec x$. We need to find the second derivative of this function.

Step 2 ：First, we rewrite the secant function in terms of cosine: $y=\frac{4}{\cos x}$.

Step 3 ：Next, we find the first derivative of $y$ using the quotient rule. The quotient rule states that the derivative of $\frac{f(x)}{g(x)}$ is $\frac{f^{\prime }(x)g(x)-f(x)g^{\prime }(x)}{[g(x){]}^2}$. Here, $f(x)=4$ and $g(x)=\cos x$. The derivative of $f(x)$ is 0 and the derivative of $g(x)$ is $-\sin x$.

Step 4 ：Applying the quotient rule, we find that the first derivative of $y$ is $y^{\prime }=\frac{4\sin x}{{\cos }^{2}x}$.

Step 5 ：We then find the second derivative by differentiating the first derivative. Again using the quotient rule, we find that the second derivative of $y$ is $y^{″}=\frac{8{\sin }^{2}x}{{\cos }^{3}x}+\frac{4}{\cos x}$.

Step 6 ：$\boxed{{\displaystyle y^{″}=\frac{8{\sin }^{2}x}{{\cos }^{3}x}+\frac{4}{\cos x}}}$ is the final answer.