Final Answer
$2x^{\left(\ln\left(x\right)-1\right)}\ln\left(x\right)$
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Step-by-step Solution
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1
To derive the function $x^{\ln\left(x\right)}$, use the method of logarithmic differentiation. First, assign the function to $y$, then take the natural logarithm of both sides of the equation
$y=x^{\ln\left(x\right)}$
2
Apply natural logarithm to both sides of the equality
$\ln\left(y\right)=\ln\left(x^{\ln\left(x\right)}\right)$
3
Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$
$\ln\left(y\right)=\ln\left(x\right)\ln\left(x\right)$
4
Derive both sides of the equality with respect to $x$
$\frac{d}{dx}\left(\ln\left(y\right)\right)=\frac{d}{dx}\left(\ln\left(x\right)\ln\left(x\right)\right)$
5
When multiplying two powers that have the same base ($\ln\left(x\right)$), you can add the exponents
$\frac{d}{dx}\left(\ln\left(y\right)\right)=\frac{d}{dx}\left(\ln\left(x\right)^2\right)$
6
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$\frac{d}{dx}\left(\ln\left(y\right)\right)=2\frac{d}{dx}\left(\ln\left(x\right)\right)\ln\left(x\right)$
Intermediate steps
7
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 $\displaystyle f'(x)=\frac{a'}{a}$
$\frac{1}{y}\frac{d}{dx}\left(y\right)=2\left(\frac{1}{x}\right)\frac{d}{dx}\left(x\right)\ln\left(x\right)$
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Intermediate steps
8
The derivative of the linear function is equal to $1$
$\frac{y^{\prime}}{y}=2\left(\frac{1}{x}\right)\frac{d}{dx}\left(x\right)\ln\left(x\right)$
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Intermediate steps
9
The derivative of the linear function is equal to $1$
$\frac{y^{\prime}}{y}=2\left(\frac{1}{x}\right)\ln\left(x\right)$
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10
Multiply the fraction and term
$\frac{y^{\prime}}{y}=\frac{2\ln\left(x\right)}{x}$
11
Multiply both sides of the equation by $y$
$y^{\prime}=\frac{2y\ln\left(x\right)}{x}$
12
Substitute $y$ for the original function: $x^{\ln\left(x\right)}$
$y^{\prime}=\frac{2x^{\ln\left(x\right)}\ln\left(x\right)}{x}$
13
Simplify the fraction $\frac{2x^{\ln\left(x\right)}\ln\left(x\right)}{x}$ by $x$
$y^{\prime}=2x^{\left(\ln\left(x\right)-1\right)}\ln\left(x\right)$
14
The derivative of the function results in
$2x^{\left(\ln\left(x\right)-1\right)}\ln\left(x\right)$
Final Answer
$2x^{\left(\ln\left(x\right)-1\right)}\ln\left(x\right)$