# Step-by-step Solution

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## Final Answer

$\left(\frac{4\left(5x^{4}+3\right)}{x^5+3x}+\frac{\sin\left(x\right)}{\cos\left(x\right)}\right)\frac{\left(x^5+3x\right)^4}{\cos\left(x\right)}$

## Step-by-step explanation

Problem to solve:

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

Choose the solving method

1

To derive the function $\frac{\left(x^5+3x\right)^4}{\cos\left(x\right)}$, use the method of logarithmic differentiation. First, assign the function to $y$, then take the natural logarithm of both sides of the equation

$y=\frac{\left(x^5+3x\right)^4}{\cos\left(x\right)}$
2

Apply logarithm to both sides of the equality

$\ln\left(y\right)=\ln\left(\frac{\left(x^5+3x\right)^4}{\cos\left(x\right)}\right)$
3

The logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator

$\ln\left(y\right)=\ln\left(\left(x^5+3x\right)^4\right)-\ln\left(\cos\left(x\right)\right)$
4

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

$\ln\left(y\right)=4\ln\left(x^5+3x\right)-\ln\left(\cos\left(x\right)\right)$
5

Derive both sides of the equality with respect to $x$

$\frac{d}{dx}\left(\ln\left(y\right)\right)=\frac{d}{dx}\left(4\ln\left(x^5+3x\right)-\ln\left(\cos\left(x\right)\right)\right)$
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 $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{y}\frac{d}{dx}\left(y\right)=\frac{d}{dx}\left(4\ln\left(x^5+3x\right)-\ln\left(\cos\left(x\right)\right)\right)$
7

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

$y^{\prime}\frac{1}{y}=\frac{d}{dx}\left(4\ln\left(x^5+3x\right)-\ln\left(\cos\left(x\right)\right)\right)$
8

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

$y^{\prime}\frac{1}{y}=\frac{d}{dx}\left(4\ln\left(x^5+3x\right)\right)+\frac{d}{dx}\left(-\ln\left(\cos\left(x\right)\right)\right)$
9

The derivative of a function multiplied by a constant ($4$) is equal to the constant times the derivative of the function

$y^{\prime}\frac{1}{y}=4\frac{d}{dx}\left(\ln\left(x^5+3x\right)\right)+\frac{d}{dx}\left(-\ln\left(\cos\left(x\right)\right)\right)$
10

The derivative of a function multiplied by a constant ($-1$) is equal to the constant times the derivative of the function

$y^{\prime}\frac{1}{y}=4\frac{d}{dx}\left(\ln\left(x^5+3x\right)\right)-\frac{d}{dx}\left(\ln\left(\cos\left(x\right)\right)\right)$
11

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

$y^{\prime}\frac{1}{y}=4\left(\frac{1}{x^5+3x}\right)\frac{d}{dx}\left(x^5+3x\right)-\frac{d}{dx}\left(\ln\left(\cos\left(x\right)\right)\right)$
12

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

$y^{\prime}\frac{1}{y}=4\left(\frac{1}{x^5+3x}\right)\frac{d}{dx}\left(x^5+3x\right)-\left(\frac{1}{\cos\left(x\right)}\right)\frac{d}{dx}\left(\cos\left(x\right)\right)$
13

The derivative of the cosine of a function is equal to minus the sine of the function times the derivative of the function, in other words, if $f(x) = \cos(x)$, then $f'(x) = -\sin(x)\cdot D_x(x)$

$y^{\prime}\frac{1}{y}=4\left(\frac{1}{x^5+3x}\right)\frac{d}{dx}\left(x^5+3x\right)+\frac{\sin\left(x\right)}{\cos\left(x\right)}$
14

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

$y^{\prime}\frac{1}{y}=4\left(\frac{1}{x^5+3x}\right)\left(\frac{d}{dx}\left(x^5\right)+\frac{d}{dx}\left(3x\right)\right)+\frac{\sin\left(x\right)}{\cos\left(x\right)}$
15

The derivative of the linear function times a constant, is equal to the constant

$y^{\prime}\frac{1}{y}=4\left(\frac{1}{x^5+3x}\right)\left(\frac{d}{dx}\left(x^5\right)+3\right)+\frac{\sin\left(x\right)}{\cos\left(x\right)}$
16

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

$y^{\prime}\frac{1}{y}=\frac{4\left(5x^{4}+3\right)}{x^5+3x}+\frac{\sin\left(x\right)}{\cos\left(x\right)}$
17

Divide both sides of the equation by $\frac{1}{y}$

$y^{\prime}=y\left(\frac{4\left(5x^{4}+3\right)}{x^5+3x}+\frac{\sin\left(x\right)}{\cos\left(x\right)}\right)$
18

Substitute $y$ for the original function: $\frac{\left(x^5+3x\right)^4}{\cos\left(x\right)}$

$y^{\prime}=\left(\frac{4\left(5x^{4}+3\right)}{x^5+3x}+\frac{\sin\left(x\right)}{\cos\left(x\right)}\right)\frac{\left(x^5+3x\right)^4}{\cos\left(x\right)}$
19

The derivative of the function results in

$\left(\frac{4\left(5x^{4}+3\right)}{x^5+3x}+\frac{\sin\left(x\right)}{\cos\left(x\right)}\right)\frac{\left(x^5+3x\right)^4}{\cos\left(x\right)}$

## Final Answer

$\left(\frac{4\left(5x^{4}+3\right)}{x^5+3x}+\frac{\sin\left(x\right)}{\cos\left(x\right)}\right)\frac{\left(x^5+3x\right)^4}{\cos\left(x\right)}$
$\frac{d}{dx}\left(\frac{\left(x^5+3x\right)^4}{cos\:x}\right)$

### Main topic:

Logarithmic differentiation

19

### Time to solve it:

~ 0.11 s (SnapXam)