Question

In Exercises \(53-60\), find the derivative of the function. \(y=\cosh ^{-1}(3 x)\)

Short answer

The derivative of the function \(y=\cosh ^{-1}(3 x)\) is \(\frac{3}{\sqrt{9x^2 - 1}}\).

Step-by-step solution

Identify the Function and the Derivative Rule

The given function is \(y=\cosh ^{-1}(3 x)\) which involves the inverse hyperbolic cosine function. The derivative of \(\cosh ^{-1}(x)\) is \(\frac{1}{\sqrt{x^2 - 1}}\), provided \(x > 1\). The chain rule must be employed since the function contains an inner function 3x.

Apply Chain Rule

According to the chain rule, \((f(g(x)))'=f'(g(x)).g'(x)\). Here, \(f(x) = \cosh ^{-1}(x) \) and \(g(x) = 3x\). Hence, the derivative of the given function would also involve the derivative of 3x, which is 3.

Take the Derivative

The derivative of the given function will be \(\frac{1}{\sqrt{(3x)^2 - 1}}*3\), or \(\frac{3}{\sqrt{9x^2 - 1}}\).