Introduction
In derivatives, there are many different rules of differentiations according to the function. The product rule and chain rule are one of those important rules that are necessary. In this article, we will discuss their differences and learn how to apply product rule step-by-step.
What is the Product and Chain Rule?
Product Rule
The product rule is a principle of differentiating a function formed by the product of two different functions. One of the primary and essential rules of derivatives follows the concepts of limits and derivatives. It is also known as Leibniz Rule.
According to this rule, if two functions are differentiable, the derivative of both functions can be calculated as their product. One notable fact about this rule is that it can be extended or generalized to the product of three more functions.
By definition, if f(x) and g(x) are two differentiable functions, the product rule for y = f(x) . g(x) can be written as;
$$\frac{dy}{dx}\;=\;f'(x).g(x)\;+\;f(x) g'(x)$$
Or,
$$\frac{dy}{dx}\;=\;\frac{df(x)}{dx}\;×\;g(x)\;+\;f(x)\;×\;\frac{dg(x)}{dx}$$
Chain Rule
The chain rule in calculus is a formula that is used to differentiate two functions combined and formed with each other. It can also differentiate the complex function and is difficult to differentiate by definition of the derivative.
In derivatives, the chain rule states that a complex function or a combination of two functions can be differentiable by substituting the inner function as u. Due to the substitution technique, the chain rule is also known as u-substitution or substitution. If two functions f(x) and g(x) can be written as the function of another one then,
$$y\;=\;f(g(x))$$
Let u = g(x) then above function will become:
$$y\;=\;f(u)$$
The chain rule formula is defined as:
$$\frac{dy}{dx}\;=\;\frac{dy}{du}\;\times\;\frac{du}{dx}$$
Note: “You can use Chain Rule Calculator to simplify the process of calculating derivatives for composite functions. With a chain rule calculator, you can input the composite function, and the calculator will apply the chain rule algorithm to compute the derivative efficiently and accurately.”
Is chain rule same as product rule?
Although the chain and product rules are essential concepts in calculus to find derivatives, both can be generalized to find derivatives of three or more functions. But, the answer is no, both are not the same. You can confirm this by discussing the comparison between both rules. Let’s discuss the comparison in the following difference table.
Due to the above differences, it is obvious that the chain and product rules are not the same.
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Chain Rule
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Product Rule
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It is a technique of finding derivatives of a function that appears in a combination of two functions.
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It is a technique of calculating derivatives of two functions in the form of their product.
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The chain rule formula for a function y = f(g(x) is,
$$\frac{dy}{dx}\;=\;\frac{dy}{du}\;\times\;\frac{du}{dx}$$
Where u = g(x)
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The product rule formula for a function y = f(x).g(x) is,
$$\frac{dy}{dx}\;=\;f'(x).g(x)\;+\;f(x)g'(x)$$
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It calculates the rate of change of a function with respect to another function.
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It calculates the rate of change of product of two functions at the same time.
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It is required to use a substitution method to find derivatives.
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It does not require the use of a substitution method, it calculates derivatives directly.
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How do you use the product rule step-by-step?
The product rule for derivatives is applicable when you have two functions in product form. For example, in y = x there are two functions such that,
$$f(x)\;=\;x\;\;,\;\;g(x)\;=\;sin\;x $$
Here you can use the following step-by-step method to apply the product rule on the above function.
- Simplify the expression, such that,
$$y\;=\;x $$
- Apply the product rule.
$$\frac{dy}{dx}\;=\;f'(x).g(x)\;+\;f(x)g'(x)$$
- Substitute the derivatives of each function in the formula.
$$\frac{dy}{dx}\;=\;1.sin\;x\;+\;x.(cos x) $$
- Simplify the expression.
$$\frac{dy}{dx}\;=\;sin\;x\;+\;xx $$
Product Rule Formulas
The product rule for two functions can be written in many versions.
- Suppose f and g are functions of x and both are differentiable at a point x = xo , then product rule for specific point x = xo is,
- $$\frac{d}{dx}|f(x)g(x)|_{x=x_o}\;=\;|\frac{df(x)}{dx}_{x=x_o}\;×\;g(x)\;+\;f(x)\;×\;|\frac{dg(x)}{dx}|_{x=x_o}$$
- Suppose f and g are functions of x, then product rule in point notation is,
- $$\frac{d}{dx}[f(x)g(x)]\;=\;f'(x).g(x)\;+\;f(x)g'(x)$$
- Suppose f and g are functions of x, then point free notation of product rule,
- $$(fg)'\;=\;f'g\;+\;fg'$$
- Suppose u and v are dependent variables on x, then the pure Leibniz Notation is,
- $$\frac{d(uv)}{dx}\;=\;\frac{du}{dx}v\;+\;u\frac{dv}{dx}$$
- Product rule in terms of differentials is,
- $$d(uv)\;=\;v(du)\;+\;u(dv)$$
FAQ’s
When should I use Product Rule?
When you have a function that is the product of two different functions of an independent variable x. Then, to differentiate it, the product rule is helpful. It is because you can’t directly differentiate both functions at the same time. Therefore, you should use product rule.
What is the Reverse Product Rule?
Same as the reverse of chain rule is the substitution method in integration, the reverse of chain rule is the partial integration rule in integrals. It is used to find the integration of products of two functions. For a function y = f(x).g(x), the reverse of product rule is,
$$ ∫\;ydy\;=\;f(x)\;∫\;g(x)dx\;-\;∫\;[\frac{d}{dx}(f(x).∫z;g(x)dx ]$$
Can you use Chain Rule for Integration?
Yes, you can use chain rule to calculate integration of a function by using a substitution method. In integrals, the chain rule is called reverse chain rule or u-substitution or substation rule. It allows you to find the antiderivative of a complex function by using the substitution method.
What is the Opposite of Power Rule?
Since the power rule is used to find the derivative of a function that is in some power. In integrals, the opposite of power rule increases the power by one and divides by power +1. It integrates a function in power.
$$∫\;x^n dx\;=\;\frac{x^{n+1}}{n+1}\;+\;c$$