Introduction
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 product of two functions. For this, the product rule formula is used to find the derivative of a product of two functions. Let’s understand how to apply the product rule to find a derivative.
Understanding the Product Rule
In derivatives, the rate of change of two functions multiplied together is known as the product rule of differentiation. It is used to find the slope of two functions at the same time. By definition, it is defined as;
“The derivative of two functions multiplied together is equal to the derivative of the first function multiplied with the second function plus the derivative of the second function multiplied with the first function.”
Product Rule Formula
If two functions f(x) and g(x) are multiplied together. Then, by the product rule formula, the derivative of f(x)g(x) can be written as:
$$\frac{d}{dx}[f(x)g(x)]=f’(x)g(x)+f(x)g’(x)$$
Where,
f’(x)= is the derivative of first function f(x)
g’(x)= is the derivative of second function g(x)
This rule is helpful for different types of functions like exponential, logarithmic function etc. If the one of the two functions in the product rule, is a constant, then the rule will be known as constant multiple rule and represented as;
$$\frac{d}{dx}[af(x)]=af’(x)$$
Where $‘a’$ is a real number. We can also define the product rule formula for different functions given below.
Product Rule formula for exponents
If two exponential functions are multiplied together, the product rule can be used to calculate derivative. For example, the derivative of the product of em and en is,
$$\frac{d}{dx}[e^m× e^n] = me^m×e^n+ne^m×e^n$$
Since in exponents, $e^m× e^n$ can be written as $e^{m+n}$. Therefore, the product rule formula for exponents is,
$$\frac{d}{dx}[e^m× e^n] = me^{m+n}+ne^{m+n}$$
Or,
$$\frac{d}{dx}[e^m× e^n] = (m+n)e^{m+n}$$
Where,
- $\frac{d}{dx}[e^m]=me^m$
- $\frac{d}{dx}[e^n]=me^n$
Product Rule for Logarithm
The derivative of the product of two logarithmic functions can be calculated by using the following formula.
$$\frac{d}{dx}[\log_a XY]=\frac{d}{dx}[\log_a X]+\frac{d}{dx}[\log_a Y]$$
Where, logaX and logaYare the logarithmic functions with the base a.
Product rule for three functions
Generally, the product rule of the derivative is defined for the multiple of two functions. But sometimes, we need to calculate the rate of change of three functions combined; then, the product rule helps to find derivatives. So, for the product of three functions u(x), v(x) and w(x), the product rule for derivative is defined as;
$$\frac{d}{dx}[u(x)v(x)w(x)]= u’(x).v(x)w(x)+u(x).v’(x).w(x)+u(x)v(x)w’(x)$$
Where,
- u’(x) is the derivative of u(x)
- v’(x) is the derivative of v(x)
- w’(x) is the derivative of w(x)
If these functions are in fractional form, then the quotient rule for three functions can be used to find rate of change.
How to apply the product rule of derivative?
The derivative of two functions at a time can be calculated by using the product rule. We can apply the product rule formula by using the following steps.
- Write the expression of the function.
- Identify the product 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 sum of the results obtained in step 5 and simplify the answer if needed.
Let’s understand the following examples by applying the product rule of derivative.
Product Rule formula example 1
To calculate the derivative of secxtanx, the product rule formula can be used as;
$$y=\sec x\tan x$$
Apply the derivative on both sides with respect to x.
$$\frac{dy}{dx}=\frac{d}{dx}[\sec x\tan x]$$
Assuming that, the first function is sec x and second function is tan x. Now apply the product rule formula,
$$\frac{dy}{dx} = \frac{d}{dx}[\sec x]\tan x + \sec x\frac{d}{dx}[\tan x]$$
Finding the derivatives of both functions,
$$\frac{dy}{dx} = [\sec x\tan x]\tan x + \sec x[\sec^2x]$$
Now simplifying,
$$\frac{dy}{dx}=\sec x\tan^2x+\sec^3x$$
More simplification,
$$\frac{dy}{dx}=\sec x[\tan^2x+\sec^2x]$$
Since $\tan^2x+\sec^2x=-1$, then
$$\frac{dy}{dx}=-\sec x$$
Product Rule formula example 2
To calculate the derivative of e squared x, the product rule of the derivative can be used.
$$y=e^{2x}$$
Applying derivative on both sides,
$$\frac{dy}{dx}=\frac{d}{dx}[e^{2x}]$$
We can write the above equation as;
$$\frac{dy}{dx}=\frac{d}{dx}[e^x.e^x]$$
Applying the product rule of derivatives,
$$\frac{dy}{dx} = e^x\frac{d}{dx}[e^x]+\frac{d}{dx}[e^x]e^x$$
Since the derivative of exponential function is itself an exponential function, then,
$$\frac{dy}{dx}=e^x.e^x+e^x.e^x$$
Or,
$$\frac{dy}{dx} = 2e^{2x}$$
Since the function y= e2x is an exponential function with a real exponent 2, we can also use the power rule formula to find its derivative.
Applying Product rule formula by using calculator
The derivative of a product of two or more functions can be also calculated by using the product rule derivative calculator. It is an online tool to differentiate products of functions by using the multiplication rule formula. You can find it online by searching for a derivative calculator. For example, to calculate the derivative of cos squared x, the following steps are used by using this calculator.
- Write the expression of the power function in the input box such as, cos^2x.
- 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 product rule calculator will provide the derivative of cos squared quickly and accurately which will be -2cosx sinx.
Comparison between product rule and chain rule
The comparison between the product and chain rules can be easily analysed using the following difference table.
Product Rule |
Chain Rule |
The product rule calculus is used to differentiate a product of two functions. |
The chain rule is used to differentiate a function which is combined with another function. |
The product rule is defined as; $\frac{d}{dx}[f(x)g(x)]=f’(x)g(x)+f(x)g’(x)$ |
The difference rule is defined as; $\frac{d}{dx}[f(g(x))]=\frac{dy}{du} × \frac{du}{dx}$ Where, y=f(u) and u=g(x) |
The product rule is used when there is a product of two or more functions. |
The chain rule is used when the function is in a combined form with another function which can be replaced by a perimeter. |
Conclusion
The product rule in derivative is a technique of finding derivative of a product of two functions. It allows us to find the rate of change in a function with respect to the rate of change in the other function. It has many applications in calculus as well as in real life. For example, we can calculate the rate of change of a quantity with respect to another one.