Question

Find the derivative. Simplify where possible.

48. \(g\left( x \right) = {\tanh ^{ - 1}}\left( {{x^3}} \right)\)

Short answer

The derivative is \(g'\left( x \right) = \frac{{3{x^2}}}{{1 - {x^6}}}\).

Step-by-step solution

Apply the derivative of composite function

On applying the derivative of the given function\(g\left( x \right) = {\tanh ^{ - 1}}\left( {{x^3}} \right)\), we get, as follows:

\(g'\left( x \right) = \frac{d}{{dx}}\left( {{{\tanh }^{ - 1}}\left( {{x^3}} \right)} \right)\frac{d}{{dx}}\left( {{x^3}} \right)\)

Plug in the derivatives and simplify

The derivative of\({\tanh ^{ - 1}}x\)is \(\frac{1}{{1 - {x^2}}}\). On plugging this derivative and simplifying we get:

\(\begin{aligned}g'\left( x \right) & = \frac{d}{{dx}}\left( {{{\tanh }^{ - 1}}\left( {{x^3}} \right)} \right)\frac{d}{{dx}}\left( {{x^3}} \right)\\ & = \frac{1}{{1 - {{\left( {{x^3}} \right)}^2}}}\left( {3{x^2}} \right)\\ & = \frac{{3{x^2}}}{{1 - {x^6}}}\end{aligned}\)

So, \(g'\left( x \right) = \frac{{3{x^2}}}{{1 - {x^6}}}\).