# Examples of Derivatives of Hyperbolic Functions

Example: Differentiate ${x^3}{\tanh ^2}\sqrt x$ with respect to $x$.

Consider the function $y = {x^3}{\tanh ^2}\sqrt x$

Differentiating both sides with respect to $x$, we have
$\frac{{dy}}{{dx}} = \frac{d}{{dx}}{x^3}{\tanh ^2}\sqrt x$

Using the product rule of differentiation, we have
$\begin{gathered} \frac{{dy}}{{dx}} = {\tanh ^2}\sqrt x \frac{d}{{dx}}{x^3} + {x^3}\frac{d}{{dx}}{\tanh ^2}\sqrt x \\ \Rightarrow \frac{{dy}}{{dx}} = {\tanh ^2}\sqrt x \left( {3{x^2}} \right) + {x^3}\left( {2\tanh \sqrt x \frac{d}{{dx}}\tanh \sqrt x } \right) \\ \Rightarrow \frac{{dy}}{{dx}} = 3{x^2}{\tanh ^2}\sqrt x + {x^3}2\tanh \sqrt x \sec {{\text{h}}^2}\sqrt x \frac{d}{{dx}}\sqrt x \\ \Rightarrow \frac{{dy}}{{dx}} = 3{x^2}{\tanh ^2}\sqrt x + 2{x^3}\tanh \sqrt x \sec {{\text{h}}^2}\sqrt x \frac{1}{{2\sqrt x }} \\ \Rightarrow \frac{{dy}}{{dx}} = 3{x^2}{\tanh ^2}\sqrt x + {x^3}\tanh \sqrt x \sec {{\text{h}}^2}\sqrt x \frac{1}{{\sqrt x }} \\ \Rightarrow \frac{{dy}}{{dx}} = 3{x^2}{\tanh ^2}\sqrt x + {x^{\frac{5}{2}}}\tanh \sqrt x \sec {{\text{h}}^2}\sqrt x \\ \end{gathered}$

Example: Find $\frac{{dy}}{{dx}}$ if the given function is $y = \sinh \left( {\cos 3x} \right)$

We have the given function
$y = \sinh \left( {\cos 3x} \right)$

Differentiating both sides with respect to $x$, we have
$\begin{gathered} \frac{{dy}}{{dx}} = \frac{d}{{dx}}\sinh \left( {\cos 3x} \right) \\ \Rightarrow \frac{{dy}}{{dx}} = \cosh \left( {\cos 3x} \right)\frac{d}{{dx}}\cos 3x \\ \Rightarrow \frac{{dy}}{{dx}} = \cosh \left( {\cos 3x} \right)\left( { – \sin 3x} \right)\frac{d}{{dx}}3x \\ \Rightarrow \frac{{dy}}{{dx}} = \cosh \left( {\cos 3x} \right)\left( { – \sin 3x} \right)\left( 3 \right) \\ \Rightarrow \frac{{dy}}{{dx}} = – 3\cosh \left( {\cos 3x} \right)\sin 3x \\ \end{gathered}$