# Step-by-step Solution

## Find the derivative using the quotient rule $\frac{d}{dx}\left(\frac{\left(x^5+3x\right)^4}{\cos\left(x\right)}\right)$

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### Videos

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

## 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

Apply the quotient rule for differentiation, which states that if $f(x)$ and $g(x)$ are functions and $h(x)$ is the function defined by $h(x) = \frac{f(x)}{g(x)}$, where ${g(x) \neq 0}$, then $h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}$

$\frac{\frac{d}{dx}\left(\left(x^5+3x\right)^4\right)\cos\left(x\right)-\left(x^5+3x\right)^4\frac{d}{dx}\left(\cos\left(x\right)\right)}{\cos\left(x\right)^2}$
2

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

$\frac{4\left(x^5+3x\right)^{\left(4-1\right)}\frac{d}{dx}\left(x^5+3x\right)\cos\left(x\right)-\left(x^5+3x\right)^4\frac{d}{dx}\left(\cos\left(x\right)\right)}{\cos\left(x\right)^2}$
3

Subtract the values $4$ and $-1$

$\frac{4\left(x^5+3x\right)^{3}\frac{d}{dx}\left(x^5+3x\right)\cos\left(x\right)-\left(x^5+3x\right)^4\frac{d}{dx}\left(\cos\left(x\right)\right)}{\cos\left(x\right)^2}$
4

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 $f'(x) = -\sin(x)\cdot D_x(x)$

$\frac{4\left(x^5+3x\right)^{3}\frac{d}{dx}\left(x^5+3x\right)\cos\left(x\right)+\left(x^5+3x\right)^4\sin\left(x\right)}{\cos\left(x\right)^2}$
5

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

$\frac{4\left(x^5+3x\right)^{3}\left(\frac{d}{dx}\left(x^5\right)+\frac{d}{dx}\left(3x\right)\right)\cos\left(x\right)+\left(x^5+3x\right)^4\sin\left(x\right)}{\cos\left(x\right)^2}$
6

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

$\frac{4\left(x^5+3x\right)^{3}\left(\frac{d}{dx}\left(x^5\right)+3\right)\cos\left(x\right)+\left(x^5+3x\right)^4\sin\left(x\right)}{\cos\left(x\right)^2}$
7

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

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

$\frac{4\left(x^5+3x\right)^{3}\left(5x^{4}+3\right)\cos\left(x\right)+\left(x^5+3x\right)^4\sin\left(x\right)}{\cos\left(x\right)^2}$
$\frac{d}{dx}\left(\frac{\left(x^5+3x\right)^4}{cos\:x}\right)$