Derive the function e^(yln(x)*x) with respect to x

\frac{d}{dx}\left(e^{xy\cdot \ln\left(x\right)}\right)

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Answer

$yx^{y\cdot x}\left(x\frac{1}{x}+\ln\left(x\right)\right)$

Step by step solution

Problem

$\frac{d}{dx}\left(e^{xy\cdot \ln\left(x\right)}\right)$
1

Applying the derivative of the exponential function

$1\frac{d}{dx}\left(y\cdot x\ln\left(x\right)\right)e^{y\cdot x\ln\left(x\right)}$
2

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$1ye^{y\cdot x\ln\left(x\right)}\cdot\frac{d}{dx}\left(x\ln\left(x\right)\right)$
3

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=x$ and $g=\ln\left(x\right)$

$1ye^{y\cdot x\ln\left(x\right)}\left(x\frac{d}{dx}\left(\ln\left(x\right)\right)+\ln\left(x\right)\frac{d}{dx}\left(x\right)\right)$
4

The derivative of the linear function is equal to $1$

$1ye^{y\cdot x\ln\left(x\right)}\left(x\frac{d}{dx}\left(\ln\left(x\right)\right)+1\ln\left(x\right)\right)$
5

The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

$1ye^{y\cdot x\ln\left(x\right)}\left(x\frac{1}{x}\cdot\frac{d}{dx}\left(x\right)+1\ln\left(x\right)\right)$
6

The derivative of the linear function is equal to $1$

$1ye^{y\cdot x\ln\left(x\right)}\left(1x\left(\frac{1}{x}\right)+1\ln\left(x\right)\right)$
7

Any expression multiplied by $1$ is equal to itself

$y\left(x\frac{1}{x}+\ln\left(x\right)\right)e^{y\cdot x\ln\left(x\right)}$
8

Using the power rule of logarithms

$y\left(x\frac{1}{x}+\ln\left(x\right)\right)e^{\ln\left(x^{y\cdot x}\right)}$
9

Simplifying the logarithm

$yx^{y\cdot x}\left(x\frac{1}{x}+\ln\left(x\right)\right)$
10

Rewriting the exponent

$y\left(x^x\right)^y\left(x\frac{1}{x}+\ln\left(x\right)\right)$
11

Applying the power of a power property

$yx^{y\cdot x}\left(x\frac{1}{x}+\ln\left(x\right)\right)$

Answer

$yx^{y\cdot x}\left(x\frac{1}{x}+\ln\left(x\right)\right)$