Question

Find the derivative of the function. \(y=\tanh ^{-1}(\sin 2 x)\)

Short answer

The derivative of \(y = \tanh^{-1}(\sin(2x))\) is \(2\cos(2x) / (1-\sin^2(2x))\).

Step-by-step solution

Identify the Inner Function

Identify the inner function inside the inverse hyperbolic tangent function. Here it is \(\sin(2x)\). Let this inner function be denoted as \(u = \sin(2x)\).

Derivative of Inner Function

Take the derivative of \(u\) with respect to \(x\). The derivative of \(\sin(2x)\) can be found using the chain rule. It would yield \(u'(x) = \cos(2x) * 2\).

Applying The Derivative of Inverse Hyperbolic Tangent Function

The derivative of the inverse hyperbolic tangent function is \(1/(1-u^2)\). Substituting \(u\) with \(\sin(2x)\), this yields \(1/(1-\sin^2(2x))\).

Calculating The Final Derivative

Use the chain rule to calculate the final derivative. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. Here, the derivative of the overall function y is \(\frac{u'(x)}{(1-\sin^2(2x))}\), which simplifies to \(2\cos(2x) / (1-\sin^2(2x))\).