Question

compute the derivative of the given function.
$k(x)=\cos^{-1}(6x)$

Explanation:

Step1: Recall the derivative formula for inverse cosine function

The derivative of \(y = \cos^{-1}(u)\) is \(y^\prime=-\frac{1}{\sqrt{1 - u^{2}}}\cdot u^\prime\) (by the chain - rule). Here \(u = 6x\), and \(u^\prime=\frac{d}{dx}(6x)=6\).

Step2: Substitute \(u\) and \(u^\prime\) into the formula

Substitute \(u = 6x\) and \(u^\prime=6\) into the formula \(k^\prime(x)=-\frac{1}{\sqrt{1-(6x)^{2}}}\cdot6\).

Answer:

\(k^\prime(x)=-\frac{6}{\sqrt{1 - 36x^{2}}}\)