Derive the function ln(2x^2+4x) with respect to x

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

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Answer

$\frac{4+4x}{4x+2x^2}$

Step by step solution

Problem

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

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

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

The derivative of a sum of two functions is the sum of the derivatives of each function

$\frac{1}{4x+2x^2}\left(\frac{d}{dx}\left(4x\right)+\frac{d}{dx}\left(2x^2\right)\right)$
3

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

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

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

$\frac{1}{4x+2x^2}\left(1\cdot 4+2\frac{d}{dx}\left(x^2\right)\right)$
5

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

$\left(1\cdot 4+2\cdot 2x\right)\frac{1}{4x+2x^2}$
6

Multiply $2$ times $2$

$\left(4+4x\right)\frac{1}{4x+2x^2}$
7

Multiplying the fraction and term

$\frac{4+4x}{4x+2x^2}$

Answer

$\frac{4+4x}{4x+2x^2}$

Problem Analysis

Main topic:

Differential calculus

Time to solve it:

0.22 seconds

Views:

113