Question

: Find the derivatives of the function

$f\left(x\right)=\sqrt{\tan h^3\left(x^5\right)}$

Short answer

$\frac{dy}{dx}=\frac{1}{2\sqrt{\tan h^3\left(x^5\right)}}.3\tan h^2\left(x^5\right).sec{h}^2\left(x^5\right).5x^4$

Step-by-step solution

Step 1:Given information

The given expression is$f\left(x\right)=\sqrt{\tan h^3\left(x^5\right)}$

Step 2:Simplification

Given

$f\left(x\right)=y=\sqrt{\tan h^3\left(x^5\right)}$

Differentiating w.r.t x

$=\frac{1}{2\sqrt{\tan h^3\left(x^5\right)}}.3\tan h^2\left(x^5\right).\frac{d}{dx}\left(\tan h\left(x^5\right)\right)$

$=\frac{1}{2\sqrt{\tan h^3\left(x^5\right)}}.3\tan h^2\left(x^5\right).sec{h}^2\left(x^5\right).\frac{d}{dx}\left(x^5\right)$

$=\frac{1}{\sqrt{\tan h^3\left(x^5\right)}}.3\tan h^2\left(x^5\right).sec{h}^2\left(x^5\right).5x^4$