Introduction to Quotient Rule
The derivative is a fundamental concept of calculus that involves the rate of the instantaneous change in a function. There are some derivative rules to calculate the rate of change of different functions like exponential, trigonometric, or logarithmic functions, etc. Sometimes, we have to deal with the quotient of two functions. For this, the quotient rule formula is used to find the derivative of the quotient of two functions. Let’s understand how to apply the quotient rule to find a derivative.
Understanding the Quotient Rule
The derivative is the rate of change of a function with respect to an independent variable. The quotient rule in the derivative is used to find the rate of change in two functions when one function divides the other. It follows the product rule formula to calculate the derivative of the quotient of two functions. If f(x) and g(x) are two functions in the form of $\frac{f(x)}{g(x)}$. Then by definition, the quotient rule states that,
“The derivative of f(x)/g(x) is equal to the quotient of the difference between f’(x)g(x) and g’(x)f(x) with the square of g(x), where f’(x) and g’(x) are the derivatives of the functions f(x) and g(x) respectively.”
Quotient Rule Formula
Since the quotient rule formula is used to calculate the rate of change of the quotient of two functions. It uses the product rule formula to find derivatives. For two functions f(x) and g(x), the quotient rule formula can be mathematically expressed as:
$\frac{d}{dx}[f(x)/g(x)] = \frac{f’(x)g(x) – f(x)g’(x)}{[g(x)]^2}$
Where,
- f’(x) is the derivative of the function f(x) in the numerator.
- g’(x) is the derivative of the function g(x) in the denominator.
The above formula is used to calculate derivative of two functions. Like the power rule of derivatives, the quotient rule also applied to any type of functions like algebraic, exponential, trigonometric etc.
Quotient rule formula for three functions
We want to calculate the derivative if there are three functions in a quotient. The quotient rule for two functions can be modified to use it for three functions. For example, if there are two functions, f(x) and g(x), in the numerator and one function, h(x), in the denominator. Then, the quotient rule for three functions is,
$\frac{d}{dx}\left(\frac{f(x)g(x)}{h(x)}\right) = \frac{f’(x)g(x)h(x) – f(x)g’(x)h(x) – f(x)g(x)h’(x)}{[h(x)]^2}$
Where,
- f’(x) is the derivative of the function f(x) in the numerator.
- g’(x) is the derivative of the function g(x) in the numerator.
- h’(x) is the derivative of the function h(x) in the denominator.
How to apply the quotient rule formula?
The implementation of the quotient rule of derivative is divided into a few steps. These steps assist us to calculate the derivative of two or more functions in fraction. These steps are:
- Write the expression of the function.
- Identify the quotient of two functions and name them as first and second function.
- Apply the derivative by using the product rule formula.
- Calculate the derivative of both functions one-by-one.
- Multiply the first function’s derivative with the second function. Similarly, multiply the second function’s derivative with the first function.
- Find the square of the second function in the denominator.
- Find the difference of the results obtained in step 5 and simplify the answer if needed.
- In the final step, divide the difference calculated in step 7 with the square of the second function and simplify.
Let’s understand the following examples by applying the quotient rule of derivative.
Quotient Rule formula example 1
To calculate the derivative of tan, we can use different techniques. The quotient formula of derivative is one of the efficient methods to find derivative of tanx. We can calculate it as;
$y=\tan x$
By using trigonometric ratios, tan can be written as the quotient of sinx and cosx. So,
$y = \frac{\sin x}{\cos x}$
Applying derivative with respect to x.
$\frac{dy}{dx} = \frac{d}{dx}\left(\frac{\sin x}{\cos x}\right)$
Since the above function is in a fractional form, we can use the quotient rule to calculate derivative of this function. Assume that sin x is the first function and cos x is the second one, then applying quotient rule,
$\frac{dy}{dx} = \frac{\cos x.\frac{d}{dx}(\sin x) – \sin x.\frac{d}{dx}(\cos x)}{\cos^2x}$
Since the derivative of sin x and cos x is cos x and – sin x respectively. Then,
$\frac{dy}{dx} = \frac{\cos x.\cos x – \sin x(-\cos x)}{\cos^2x}$
Or,
$\frac{dy}{dx} = \frac{\cos^2x + \sin^2x}{\cos^2x}$
Since $\cos^2x + \sin^2x = 1$ then,
$\frac{dy}{dx} = \frac{1}{\cos^2x}$
In trigonometry, the reciprocal of cos x is equal to sec x. Therefore,
$\frac{dy}{dx} = \sec^2x$
Hence the derivative of tan x is sec square x. The quotient rule also helps to calculate the derivative tan(2x) which is equal to 2sec^{2}x.
Quotient Rule Formula example 2
To calculate the derivative of cot square x, we will write it as;
$y = \cot^2x$
By using trigonometric ratios, cot x can be written as;
$y = \frac{\cos^2x}{\sin^2x}$
Applying derivative with respect to x.
$\frac{dy}{dx} = \frac{d}{dx}\left(\frac{\cos^2x}{\sin^2x}\right)$
Since the above function is a quotient of two functions cosine and sine, therefore we can use the quotient formula of derivatives.
$\frac{dy}{dx} = \frac{\sin^2x.\frac{d}{dx}(\cos^2x) – \cos^2x.\frac{d}{dx}(\sin^2x)}{\sin^4x}$
Finding the derivative of sin and cos one by one,
$\frac{dy}{dx}=\frac{\sin^2x(-2\cos x\sin x)-\cos^2x(2\sin x\cos x)}{\sin^4x}$
$\frac{dy}{dx}=\frac{-2\sin^3x\cos x – 2\sin x\cos^3x}{\sin^4x}$
Taking common -2sinx cosx.
$\frac{dy}{dx} = \frac{-2\sin x\cos x(\sin^2x + \cos^2x)}{\sin^4x}$
Since $\sin^2x + \cos^2x$ is equal to 1.
$\frac{dy}{dx} = \frac{-2\sin x \cos x}{\sin^4x}$
Or,
$\frac{dy}{dx} = \frac{-2\cos x}{\sin^3x}$
By using trigonometric ratios,
$\frac{\cos x}{\sin x}= \cot x$ and $\frac{1}{\sin^2x} = \csc^2x$ then,
$\frac{dy}{dx} = -2\cot x\csc^2x$
Hence the derivative of cot^2x is negative of the $2\cot x\csc^2x$.
Applying Quotient rule formula by using calculator
The derivative of a quotient of two or more functions can be also calculated by using the quotient rule derivative calculator. It is an online tool that follows the quotient derivative formula to find derivative. You can find it online by searching for a derivative calculator. For example, to calculate the derivative of tan 3x, the following steps are used by using this calculator.
- Write the expression of the power function in the input box such as, tan3x.
- Choose the variable to calculate the rate of change, which will be x in this example.
- Review the input so that there will be no syntax error in the function.
- Now at the last step, click on the calculate button. By using this step, the quotient rule calculator will provide the derivative of tan3x quickly and accurately which will be $3\sec^2(3x)$.
Comparison between Product rule and Quotient rule
The comparison between the product and quotient rules can be easily analysed using the following difference table.
Product Rule | Quotient Rule |
The product rule calculus is used to differentiate a product of two functions. | The quotient rule is used to calculate derivative of quotient of two functions. |
The product rule is defined as; $\frac{d}{dx}[f(x)g(x)]=f’(x)g(x)+f(x)g’(x)$ | The quotient rule is defined as; $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)= \frac{f’(x)g(x) – f(x)g’(x)}{[g(x)]^2}$ |
The product rule is used when there is a product of two or more functions. | The quotient rule is reliable when a function is divided with another. |
Conclusion
The quotient rule is a rule of differentiation which is used to calculate derivative of the quotient of two functions. It has many applications in calculus as well as in real life where we are required to calculate change in different quantities.