Differentiate $f(x)=\cos ^{-1}(7 x)$. Use exact values.
\[
f^{\prime}(x)=
\]

Step 1 ：Given the function \(f(x)=\cos ^{-1}(7 x)\).

Step 2 ：We need to find its derivative \(f^{\prime}(x)\).

Step 3 ：The derivative of the inverse cosine function is \(-\frac{1}{\sqrt{1-x^2}}\).

Step 4 ：However, we have a composite function here, so we need to use the chain rule.

Step 5 ：The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

Step 6 ：In this case, the outer function is \(\cos^{-1}(x)\) and the inner function is \(7x\).

Step 7 ：So, we need to differentiate both of these and multiply them together.

Step 8 ：The derivative of \(\cos^{-1}(x)\) is \(-\frac{1}{\sqrt{1-x^2}}\) and the derivative of \(7x\) is \(7\).

Step 9 ：Multiplying these together, we get \(-\frac{7}{\sqrt{1-49x^2}}\).

Step 10 ：Thus, the derivative of \(f(x)=\cos ^{-1}(7 x)\) is \(f^{\prime}(x)=-\frac{7}{\sqrt{1-49x^2}}\).

Step 11 ：\(\boxed{f^{\prime}(x)=-\frac{7}{\sqrt{1-49x^2}}}\)