# Derive the function ln(x)x^2 with respect to x

## \ln\left(x\right)x^2=\frac{d}{dx}\left(\ln\left(x\right)x^2\right)

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$x^2\ln\left(x\right)=2x\ln\left(x\right)+x^2\frac{1}{x}$

## Step by step solution

Problem

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

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=\ln\left(x\right)$ and $g=x^2$

$x^2\ln\left(x\right)=\frac{d}{dx}\left(x^2\right)\ln\left(x\right)+x^2\frac{d}{dx}\left(\ln\left(x\right)\right)$
2

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$x^2\ln\left(x\right)=2x\ln\left(x\right)+x^2\frac{d}{dx}\left(\ln\left(x\right)\right)$
3

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 $\displaystyle f'(x)=\frac{a'}{a}$

$x^2\ln\left(x\right)=2x\ln\left(x\right)+x^2\frac{1}{x}\cdot\frac{d}{dx}\left(x\right)$
4

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

$x^2\ln\left(x\right)=2x\ln\left(x\right)+1x^2\left(\frac{1}{x}\right)$
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Any expression multiplied by $1$ is equal to itself

$x^2\ln\left(x\right)=2x\ln\left(x\right)+x^2\frac{1}{x}$
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Using the power rule of logarithms

$x^2\ln\left(x\right)=\ln\left(x^{2x}\right)+x^2\frac{1}{x}$
7

Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$

$x^2\ln\left(x\right)=2x\ln\left(x\right)+x^2\frac{1}{x}$

$x^2\ln\left(x\right)=2x\ln\left(x\right)+x^2\frac{1}{x}$

### Main topic:

Differential calculus

0.22 seconds

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