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

## Step-by-step Solution

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###  Solution

$\frac{4\left(x^5+3x\right)^{3}\left(5x^{4}+3\right)\cos\left(x\right)+\left(x^5+3x\right)^4\sin\left(x\right)}{\cos\left(x\right)^2}$
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##  Step-by-step Solution 

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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 ${\displaystyle h(x) = \frac{f(x)}{g(x)}}$, where ${g(x) \neq 0}$, then ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

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

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

Learn how to solve problems step by step online. Find the derivative using the quotient rule d/dx(((x^5+3x)^4)/cos(x)). 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 {\displaystyle h(x) = \frac{f(x)}{g(x)}}, where {g(x) \neq 0}, then {\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^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}. 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). Any expression multiplied by 1 is equal to itself.

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

##  Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Find the derivativeFind derivative of ((x^5+3x)^4)/cosx using the product ruleFind derivative of ((x^5+3x)^4)/cosx using logarithmic differentiationFind derivative of ((x^5+3x)^4)/cosx using the definition

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a
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u
v
w
x
y
z
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(◻)
+
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×
◻/◻
/
÷
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e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch