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# Find the derivative of x^4ln(x^4)*9+ln(x)^5

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## Answer

$\frac{5}{x}\ln\left(x\right)^{4}+9\left(x^{7}\cdot\frac{4}{x^4}+16x^{3}\ln\left(x\right)\right)$

## Step by step solution

Problem

$\frac{d}{dx}\left(9x^4\cdot\ln\left(x^4\right)+\ln\left(x\right)^5\right)$
1

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

$\frac{d}{dx}\left(\ln\left(x\right)^5\right)+\frac{d}{dx}\left(9x^4\ln\left(x^4\right)\right)$
2

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

$\frac{d}{dx}\left(\ln\left(x\right)^5\right)+9\frac{d}{dx}\left(x^4\ln\left(x^4\right)\right)$
3

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=x^4$ and $g=\ln\left(x^4\right)$

$\frac{d}{dx}\left(\ln\left(x\right)^5\right)+9\left(x^4\frac{d}{dx}\left(\ln\left(x^4\right)\right)+\ln\left(x^4\right)\frac{d}{dx}\left(x^4\right)\right)$
4

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

$\frac{d}{dx}\left(\ln\left(x\right)^5\right)+9\left(x^4\frac{d}{dx}\left(\ln\left(x^4\right)\right)+4x^{3}\ln\left(x^4\right)\right)$
5

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

$5\frac{d}{dx}\left(\ln\left(x\right)\right)\ln\left(x\right)^{4}+9\left(x^4\frac{d}{dx}\left(\ln\left(x^4\right)\right)+4x^{3}\ln\left(x^4\right)\right)$
6

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

$5\left(\frac{1}{x}\right)\ln\left(x\right)^{4}\cdot\frac{d}{dx}\left(x\right)+9\left(x^4\frac{d}{dx}\left(\ln\left(x^4\right)\right)+4x^{3}\ln\left(x^4\right)\right)$
7

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

$5\cdot 1\left(\frac{1}{x}\right)\ln\left(x\right)^{4}+9\left(x^4\frac{d}{dx}\left(\ln\left(x^4\right)\right)+4x^{3}\ln\left(x^4\right)\right)$
8

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

$5\cdot 1\left(\frac{1}{x}\right)\ln\left(x\right)^{4}+9\left(x^4\frac{1}{x^4}\cdot\frac{d}{dx}\left(x^4\right)+4x^{3}\ln\left(x^4\right)\right)$
9

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

$5\cdot 1\left(\frac{1}{x}\right)\ln\left(x\right)^{4}+9\left(4x^4\frac{1}{x^4}x^{3}+4x^{3}\ln\left(x^4\right)\right)$
10

Multiply $1$ times $5$

$5\frac{1}{x}\ln\left(x\right)^{4}+9\left(4x^4\frac{1}{x^4}x^{3}+4x^{3}\ln\left(x^4\right)\right)$
11

When multiplying exponents with same base we can add the exponents

$5\frac{1}{x}\ln\left(x\right)^{4}+9\left(4\frac{1}{x^4}x^{7}+4x^{3}\ln\left(x^4\right)\right)$
12

Apply the formula: $a\frac{1}{x}$$=\frac{a}{x}$, where $a=4$ and $x=x^4$

$\frac{5}{x}\ln\left(x\right)^{4}+9\left(x^{7}\cdot\frac{4}{x^4}+4x^{3}\ln\left(x^4\right)\right)$
13

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

$\frac{5}{x}\ln\left(x\right)^{4}+9\left(x^{7}\cdot\frac{4}{x^4}+4\cdot 4x^{3}\ln\left(x\right)\right)$
14

Multiply $4$ times $4$

$\frac{5}{x}\ln\left(x\right)^{4}+9\left(x^{7}\cdot\frac{4}{x^4}+16x^{3}\ln\left(x\right)\right)$
15

Using the power rule of logarithms

$\frac{5}{x}\ln\left(x\right)^{4}+9\left(x^{7}\cdot\frac{4}{x^4}+\ln\left(x^{16x^{3}}\right)\right)$
16

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

$\frac{5}{x}\ln\left(x\right)^{4}+9\left(x^{7}\cdot\frac{4}{x^4}+16x^{3}\ln\left(x\right)\right)$

## Answer

$\frac{5}{x}\ln\left(x\right)^{4}+9\left(x^{7}\cdot\frac{4}{x^4}+16x^{3}\ln\left(x\right)\right)$

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### Main topic:

Differential calculus

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