Logarithmic Differentiation

If $y={\log }_e{\left(x^{e^x·a^y}\right)}^{y^x}$, then $\frac{dy}{dx}$ is equal to

If $x+y=3e^2$, then $D\left(x^y\right)$ vanishes when $x$ is $\lambda e^2$. Then, the value of $\lambda$ is equal to:

If $x+y=3e^2$, then $D\left(x^y\right)$ vanishes when $x$ is $\lambda e^2$. Then, the value of $\lambda$ is equal to:

If $y=\log \left(\sin \right(x^2\left)\right),\mathrm{ }0<x<\frac{\mathrm{\pi }}{2}$, and $\frac{dy}{dx}$ at $x=\frac{\sqrt{\pi }}{2}$ is $k\sqrt{\pi }$, then value of $k$ is

If $y=\sqrt{\log x+\sqrt{\log x+\sqrt{\log x+\dots .\infty }}}$, then value of $\frac{dy}{dx}$ is

If $y=x^x$, find $\frac{dy}{dx}$.

Differentiate the following w.r.t. $x$

${\left[{\left(\tan x\right)}^{\tan x}\right]}^{\tan x}$ at $x=\frac{\mathrm{\pi }}{4}$

Differentiate the following w.r.t. $x$

$e^{\log \left(\log x\right)}.\log 3x$

Differentiate the following w.r.t. $x$

${10}^{x^x}+x^{x^{10}}{x}^{{10}^x}$

Differentiate the following w.r.t. $x$

${\left(\sin x\right)}^x+\log x+{\left(\log x\right)}^{\tan x}+x^a$

Differentiate the following function w.r.t. $x$

${\left(2x\right)}^{\sec x}+{\left(\sec x\right)}^{2x}$

Differentiate the following function w.r.t. $x$

$$${\left(\sin x\right)}^{\tan x}+{\left(\cos x\right)}^{\sec x}$

Differentiate the following function w.r.t. $x$

$$${\left(\log x\right)}^x-x^{\log x}$

Differentiate the following function w.r.t. $x$

$e^{\tan x}+{\left(\log x\right)}^{\tan x}$

Differentiate the following function w.r.t. $x$

$x^x+a^x+x^a+a^a$

Differentiate the following function w.r.t.$x$

$x^x+a^a$

Differentiate the following function w.r.t. $x$

$e^{x^x}$

Differentiate the following function w.r.t. $x$

$x^{e^x}$

Differentiate the following function w.r.t. $x$

$x^{x^x}$

Differentiate the following function w.r.t. $x$

${\left(\sin x-\cos x\right)}^{\sin x+\cos x}$