Derive the function x^2x^5ln(x+2)*-2 with respect to x

\frac{d}{dx}\left(x^2-2x^5\ln\left(x+2\right)\right)

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Answer

$2x-2\left(x^5\frac{1}{2+x}+5x^{4}\ln\left(2+x\right)\right)$

Step by step solution

Problem

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

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

$\frac{d}{dx}\left(-2x^5\ln\left(2+x\right)\right)+\frac{d}{dx}\left(x^2\right)$
2

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

$\frac{d}{dx}\left(x^2\right)-2\frac{d}{dx}\left(x^5\ln\left(2+x\right)\right)$
3

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=x^5$ and $g=\ln\left(2+x\right)$

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

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

$2x-2\left(x^5\frac{d}{dx}\left(\ln\left(2+x\right)\right)+5x^{\left(5-1\right)}\ln\left(2+x\right)\right)$
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 $\displaystyle f'(x)=\frac{a'}{a}$

$2x-2\left(x^5\frac{1}{2+x}\cdot\frac{d}{dx}\left(2+x\right)+5x^{\left(5-1\right)}\ln\left(2+x\right)\right)$
6

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

$2x-2\left(x^5\frac{1}{2+x}\left(\frac{d}{dx}\left(2\right)+\frac{d}{dx}\left(x\right)\right)+5x^{\left(5-1\right)}\ln\left(2+x\right)\right)$
7

The derivative of the constant function is equal to zero

$2x-2\left(x^5\frac{1}{2+x}\left(0+\frac{d}{dx}\left(x\right)\right)+5x^{\left(5-1\right)}\ln\left(2+x\right)\right)$
8

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

$2x-2\left(\left(0+1\right)x^5\left(\frac{1}{2+x}\right)+5x^{\left(5-1\right)}\ln\left(2+x\right)\right)$
9

Subtract the values $5$ and $-1$

$2x-2\left(\left(0+1\right)x^5\left(\frac{1}{2+x}\right)+5x^{4}\ln\left(2+x\right)\right)$
10

Add the values $1$ and $0$

$2x-2\left(1x^5\left(\frac{1}{2+x}\right)+5x^{4}\ln\left(2+x\right)\right)$
11

Any expression multiplied by $1$ is equal to itself

$2x-2\left(x^5\frac{1}{2+x}+5x^{4}\ln\left(2+x\right)\right)$
12

Using the power rule of logarithms

$2x-2\left(x^5\frac{1}{2+x}+\ln\left(\left(2+x\right)^{5x^{4}}\right)\right)$
13

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

$2x-2\left(x^5\frac{1}{2+x}+5x^{4}\ln\left(2+x\right)\right)$

Answer

$2x-2\left(x^5\frac{1}{2+x}+5x^{4}\ln\left(2+x\right)\right)$

Problem Analysis

Main topic:

Differential calculus

Time to solve it:

0.33 seconds

Views:

101