Question

Find the derivative of the function. \(y=\sinh ^{-1}(\tan x)\)

Short answer

The derivative of \( y= \sinh^{-1}(\tan x) \) is \( y'= \frac{\sec^{2}(x)}{\sqrt{1+ \tan^{2}(x)}}\).

Step-by-step solution

Find the derivative of the outer function

Apply the formula for the derivative of the inverse hyperbolic sine function, \(f'(x)=(1/ \sqrt{1+x^{2}})*g'(x)\). Where, x is the argument of the sin function which in this case is \(\tan x\) and g'(x) is its derivative.

Find the derivative of the inner function

The derivative of \( \tan x \) is \( \sec^{2}(x) \).

Apply the chain rule

The chain rule states that the derivative of a composite function is the derivative of the outer function at the inner function times the derivative of the inner function. Therefore, the derivative of \( y \) is equal to \( \frac{\sec^{2}(x)}{\sqrt{1+ \tan^{2}(x)}}\).