Question

Find the derivative. Simplify where possible.

37.\(h\left( x \right) = \sinh \left( {{x^2}} \right)\)

Short answer

The derivative is \(h'\left( x \right) = 2x\cosh \left( {{x^2}} \right)\).

Step-by-step solution

Apply the derivative of composite functions

On applying the derivative to the given function\(h\left( x \right) = \sinh \left( {{x^2}} \right)\), we get, as follows:

^.\(h'\left( x \right) = \frac{d}{{dx}}\left( {\sinh {x^2}} \right)\frac{d}{{dx}}\left( {{x^2}} \right)\)

Plug in the derivatives and simplify

The derivative of\(\sinh {x^2}\)is \(\cosh {x^2}\) and \({x^2}\)is \(2x\). On plugging these derivatives and simplifying we get:

\(\begin{aligned}h'\left( x \right) &= \cosh {x^2}\left( {2x} \right)\\ &= 2x\cosh {x^2}\end{aligned}\)

Thus, the required answer is:

\(h'\left( x \right) = 2x\cosh \left( {{x^2}} \right)\)