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# Find the derivative $\frac{d}{dx}\left(\frac{x^3}{\sin\left(x\right)}\right)$

## Step-by-step Solution

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###  Solution

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

Problem to solve:

$\frac{d}{dx}\left(\frac{x^3}{\sin\left(x\right)}\right)$

Specify 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 $h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}$

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

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

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

Learn how to solve problems step by step online.

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

Learn how to solve problems step by step online. Find the derivative d/dx((x^3)/(sin(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 sine of a function is equal to the cosine of that function times the derivative of that function, in other words, if {f(x) = \sin(x)}, then {f'(x) = \cos(x)\cdot D_x(x)}.

$\frac{3x^{2}\sin\left(x\right)-x^3\cos\left(x\right)}{\sin\left(x\right)^2}$

##  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 d/dx((x^3)/(sin(x)) using the product ruleFind d/dx((x^3)/(sin(x)) using the quotient ruleFind d/dx((x^3)/(sin(x)) using logarithmic differentiationFind d/dx((x^3)/(sin(x)) using the definition
SnapXam A2

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7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

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