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# Quotient rule Calculator

## Get detailed solutions to your math problems with our Quotient rule step by step calculator. Sharpen your math skills and learn step by step with our math solver. Check out more online calculators here.

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### Difficult Problems

1

Solved example of Quotient rule

$\lim_{x\to0}\left(\frac{3x^2}{1-\cos\left(2\frac{x}{3}\right)}\right)$
2

Simplify the fraction

$\lim_{x\to0}\left(\frac{3x^2}{1-\cos\left(\frac{2}{3}x\right)}\right)$
3

Multiplying the fraction and term

$\lim_{x\to0}\left(\frac{3x^2}{1-\cos\left(\frac{2x}{3}\right)}\right)$
4

As the limit results in indeterminate form, we can apply L'Hôpital's rule

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(3x^2\right)}{\frac{d}{dx}\left(1-\cos\left(\frac{2x}{3}\right)\right)}\right)$
5

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

$\lim_{x\to0}\left(\frac{3\frac{d}{dx}\left(x^2\right)}{\frac{d}{dx}\left(1-\cos\left(\frac{2x}{3}\right)\right)}\right)$
6

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

$\lim_{x\to0}\left(\frac{6x}{\frac{d}{dx}\left(1-\cos\left(\frac{2x}{3}\right)\right)}\right)$
7

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

$\lim_{x\to0}\left(\frac{6x}{\frac{d}{dx}\left(1\right)+\frac{d}{dx}\left(-\cos\left(\frac{2x}{3}\right)\right)}\right)$
8

The derivative of the constant function is equal to zero

$\lim_{x\to0}\left(\frac{6x}{\frac{d}{dx}\left(-\cos\left(\frac{2x}{3}\right)\right)}\right)$
9

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

$\lim_{x\to0}\left(\frac{6x}{-\frac{d}{dx}\left(\cos\left(\frac{2x}{3}\right)\right)}\right)$
10

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

$\lim_{x\to0}\left(\frac{6x}{\frac{d}{dx}\left(\frac{2x}{3}\right)\sin\left(\frac{2x}{3}\right)}\right)$
11

Applying the quotient rule 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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

$\lim_{x\to0}\left(\frac{6x}{\frac{3\frac{d}{dx}\left(2x\right)-2x\frac{d}{dx}\left(3\right)}{9}\sin\left(\frac{2x}{3}\right)}\right)$
12

The derivative of the constant function is equal to zero

$\lim_{x\to0}\left(\frac{6x}{\frac{3\frac{d}{dx}\left(2x\right)+0x}{9}\sin\left(\frac{2x}{3}\right)}\right)$
13

Any expression multiplied by $0$ is equal to $0$

$\lim_{x\to0}\left(\frac{6x}{\frac{3\frac{d}{dx}\left(2x\right)+0}{9}\sin\left(\frac{2x}{3}\right)}\right)$
14

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

$\lim_{x\to0}\left(\frac{6x}{\frac{6+0}{9}\sin\left(\frac{2x}{3}\right)}\right)$
15

Add the values $6$ and $0$

$\lim_{x\to0}\left(\frac{6x}{\frac{6}{9}\sin\left(\frac{2x}{3}\right)}\right)$
16

Divide $6$ by $9$

$\lim_{x\to0}\left(\frac{6x}{\frac{2}{3}\sin\left(\frac{2x}{3}\right)}\right)$
17

As the limit results in indeterminate form, we can apply L'Hôpital's rule

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(6x\right)}{\frac{d}{dx}\left(\frac{2}{3}\sin\left(\frac{2x}{3}\right)\right)}\right)$
18

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

$\lim_{x\to0}\left(\frac{6}{\frac{d}{dx}\left(\frac{2}{3}\sin\left(\frac{2x}{3}\right)\right)}\right)$
19

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

$\lim_{x\to0}\left(\frac{6}{\frac{2}{3}\cdot\frac{d}{dx}\left(\sin\left(\frac{2x}{3}\right)\right)}\right)$
20

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 ${f'(x) = \cos(x)\cdot D_x(x)}$

$\lim_{x\to0}\left(\frac{6}{\frac{2}{3}\cos\left(\frac{2x}{3}\right)\frac{d}{dx}\left(\frac{2x}{3}\right)}\right)$
21

Applying the quotient rule 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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

$\lim_{x\to0}\left(\frac{6}{\frac{2}{3}\cos\left(\frac{2x}{3}\right)\left(\frac{3\frac{d}{dx}\left(2x\right)-2x\frac{d}{dx}\left(3\right)}{9}\right)}\right)$
22

The derivative of the constant function is equal to zero

$\lim_{x\to0}\left(\frac{6}{\frac{2}{3}\cos\left(\frac{2x}{3}\right)\left(\frac{3\frac{d}{dx}\left(2x\right)+0x}{9}\right)}\right)$
23

Any expression multiplied by $0$ is equal to $0$

$\lim_{x\to0}\left(\frac{6}{\frac{2}{3}\cos\left(\frac{2x}{3}\right)\left(\frac{3\frac{d}{dx}\left(2x\right)+0}{9}\right)}\right)$
24

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

$\lim_{x\to0}\left(\frac{6}{\frac{2}{3}\left(\frac{6+0}{9}\right)\cos\left(\frac{2x}{3}\right)}\right)$
25

Add the values $6$ and $0$

$\lim_{x\to0}\left(\frac{6}{\frac{2}{3}\left(\frac{6}{9}\right)\cos\left(\frac{2x}{3}\right)}\right)$
26

Divide $6$ by $9$

$\lim_{x\to0}\left(\frac{6}{\frac{2}{3}\cdot\frac{2}{3}\cos\left(\frac{2x}{3}\right)}\right)$
27

Apply the formula: $\frac{a}{bx}$$=\frac{\frac{a}{b}}{x}, where a=6, b=\frac{2}{3} and x=\frac{2}{3}\cos\left(\frac{2x}{3}\right) \lim_{x\to0}\left(\frac{9}{\frac{2}{3}\cos\left(\frac{2x}{3}\right)}\right) 28 Apply the formula: \frac{a}{bx}$$=\frac{\frac{a}{b}}{x}$, where $a=9$, $b=\frac{2}{3}$ and $x=\cos\left(\frac{2x}{3}\right)$

$\lim_{x\to0}\left(\frac{\frac{27}{2}}{\cos\left(\frac{2x}{3}\right)}\right)$
29

Evaluating the limit when $x$ tends to $0$

$\frac{\frac{27}{2}}{\cos\left(\frac{0\cdot 2}{3}\right)}$
30

Simplifying

$\frac{\frac{27}{2}}{\cos\left(\frac{0}{3}\right)}$