Question

$\large \displaystyle \sf \red{ \sqrt{ \frac{ { \sin}^{ - 1}(logx) + {e}^{log( {x \frac{ - 1}{ \sqrt{2} }) }^{ \frac{}{ } } } }{log( {e}^{ \frac{1}{ \sqrt{x} } } ) - {sec}^{ - 1}( \sqrt{ {x}^{3} } ) } } }$​

Answer 1

You can refer to the above attachment for solution!

Answer 2

To find the derivative of the given expression, we will first simplify it using algebraic manipulations and inverse trigonometric identities:

$\begin{aligned} \sqrt{\frac{\arcsin(\log x) + e^{\log\left(x\frac{-1}{\sqrt{2}}\right)}}{\log(e^{1/\sqrt{x}})-\operatorname{arcsec}(\sqrt{x^3})}} &= \sqrt{\frac{\arcsin(\log x) + e^{\log\left(x\frac{-1}{\sqrt{2}}\right)}}{\frac{1}{\sqrt{x}}-\operatorname{arcsec}(\sqrt{x^3})}} \\ &= \sqrt{\frac{\arcsin(\log x) + e^{\log\left(x\frac{-1}{\sqrt{2}}\right)}}{\frac{1}{\sqrt{x}}-\operatorname{arccos}(\sqrt{1-x^6})}} \end{aligned}$

Now, we can use the chain rule of differentiation to find the derivative of the expression. Let

$u = \arcsin(\log x) + e^{\log\left(x\frac{-1}{\sqrt{2}}\right)}$

and

$v = \frac{1}{\sqrt{x}}-\operatorname{arccos}(\sqrt{1-x^6})$

Then, the given expression is equal to $\sqrt{\frac{u}{v}}$. Using the chain rule, we have:

$\begin{aligned} \frac{d}{dx}\sqrt{\frac{u}{v}} &= \frac{1}{2\sqrt{\frac{u}{v}}}\cdot\frac{d}{dx}\frac{u}{v} \\ &= \frac{1}{2\sqrt{\frac{u}{v}}}\cdot\left(\frac{d}{dx}u\cdot\frac{1}{v} - u\cdot\frac{d}{dx}v\cdot\frac{1}{v^2}\right) \\ &= \frac{1}{2\sqrt{\frac{u}{v}}}\cdot\left(\frac{d}{dx}\left[\arcsin(\log x) + e^{\log\left(x\frac{-1}{\sqrt{2}}\right)}\right]\cdot\frac{1}{\frac{1}{\sqrt{x}}-\operatorname{arccos}(\sqrt{1-x^6})} - \left[\arcsin(\log x) + e^{\log\left(x\frac{-1}{\sqrt{2}}\right)}\right]\cdot\frac{d}{dx}\left[\frac{1}{\frac{1}{\sqrt{x}}-\operatorname{arccos}(\sqrt{1-x^6})}\right]\cdot\frac{1}{\left(\frac{1}{\sqrt{x}}-\operatorname{arccos}(\sqrt{1-x^6})\right)^2}\right) \\ &= \frac{1}{2\sqrt{\frac{u}{v}}}\cdot\left(\frac{1}{x\sqrt{1-(\log x)^2}} + e^{\log\left(x\frac{-1}{\sqrt{2}}\right)}\cdot\frac{\sqrt{2}}{2x(x-1)} - \frac{3x^5}{2\sqrt{1-x^6}\left(\frac{1}{\sqrt{x}}-\operatorname{arccos}(\sqrt{1-x^6})\right)^2}\right) \end{aligned}$

Therefore, the derivative of the given expression is:

$\frac{1}{2\sqrt{\frac{\arcsin(\log x) + e^{\log\left(x\frac{-1}{\sqrt{2}}\right)}}{\frac{1}{\sqrt{x}}-\operatorname{arccos}(\sqrt{1-x^6})}}}\cdot\left(\frac{1}{x\sqrt{1-(\log x)^2}} + e^{\log\left(x\frac{-1}{\sqrt{2}}\right)}\cdot\frac{\sqrt{2}}{2x(x-1)} - \frac{3x^5}{2\sqrt{1-x^6}\left(\frac{1}{\sqrt{x}}-\operatorname{arccos}(\sqrt{1-x^6})\right)^2}\right)$