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

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

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Answer

$\frac{2\ln\left(x\right)}{x}$

Step by step solution

Problem

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

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

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

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

$2\left(\frac{1}{x}\right)\ln\left(x\right)\frac{d}{dx}\left(x\right)$
3

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

$2\cdot 1\left(\frac{1}{x}\right)\ln\left(x\right)$
4

Multiply $1$ times $2$

$2\frac{1}{x}\ln\left(x\right)$
5

Apply the formula: $a\frac{1}{x}$$=\frac{a}{x}$, where $a=2$

$\frac{2}{x}\ln\left(x\right)$
6

Multiplying the fraction and term

$\frac{2\ln\left(x\right)}{x}$
7

Using the power rule of logarithms

$\frac{\ln\left(x^{2}\right)}{x}$
8

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

$\frac{2\ln\left(x\right)}{x}$

Answer

$\frac{2\ln\left(x\right)}{x}$

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Problem Analysis

Main topic:

Differential calculus

Time to solve it:

0.22 seconds

Views:

124