What is the difference between the Chain Rule and the Product Rule?


The derivative is a fundamental branch of calculus that measures the rate of change in a function by changing an independent variable. Some rules of differentiation are necessary to be used while finding derivatives. The chain rule and product rule are one of those rules. This article will teach you the difference between these two rules and when to use the chain and product rules.

What is the chain rule?

The chain rule is derived by Gottfried Wilhelm Leibniz. In calculus, the chain rule is a formula of derivative that allows you to calculate the rate of change of two functions combined together. It provides us with a technique to differentiate two functions at a time.

The chain rule is applicable to composite functions i.e. a function that is formed by combining two functions. Therefore, it is also known as the composite function rule. In this rule, two functions f(x) and g(x) are in combined form. It means that g(x) is a function of f and both are the functions of x.

The Concepts of Chain Rule

To calculate the composition of a function, the chain rule gives us a new path. It is a little bit different and risky to apply correctly sometimes, especially for sins, cos, and tan. They may apply a chain rule multiple times in one question. With a complex expression, the chain rule might have to be applied several times, making it difficult to apply correctly. It is possible to understand the chain rule quite easily, however.

Using our already learned differentiation techniques, we can differentiate a function that is composed of two or more functions. It can be cumbersome to break down a function into simpler parts which can be differentiated using all of those techniques. To calculate the derivative of a composite function, we use the chain rule, which states that the derivative of an outer function times the derivative of an inner function equals the derivative of a composite function.

The chain rule of linear equations:

A linear approximation to a function’s derivative is a tangent line to its graph. By examining linear functions, we can often gain intuition about the derivative’s properties. Linear functions, in particular, are very simple to analyze using the chain rule. With linear functions, one important subtlety of the chain rule is absent, so they are a good starting point for understanding it.

What is the Product rule?

In calculus, the product rule is defined as the product of two functions. Advanced technology-based tools provide a technique to find the rate of change of a function that is a product of two or more functions. The rule may be extended or generalized to products of three or more functions or to a rule for higher-order derivatives of a product.

This rule is applicable only for products of two functions i.e. a combination of two functions multiplied together. Therefore, it is also known as the Leibniz Product rule.

In order to differentiate products with two (or more) functions, there is a special rule, the product rule. A good example of this rule can be found in this unit. For these techniques to become second nature, you must practice them repeatedly and do plenty of practice exercises.

  • The product rule should be stated
  • Function-based products can be differentiated

Difference between Chain Rule and Product Rule

The chain rule and product rule both are defined for calculating the rate of change. These rules are often used together. They both help to differentiate two functions without using the long process of applying the derivative first principle with the help of a derivative calculator. Although along with the common concept of differentiation, these rules also differ from each other. Let’s discuss the difference between the chain rule and the product rule.

The derivative of a composite function cannot be easily calculated, i.e. you have to calculate the derivatives of both functions separately. In the product rule, the derivative of both functions can be directly calculated. The chain rule formula is applicable for composite functions but it can also be used to differentiate a product of two functions. The product rule is applicable for products of two functions only. It cannot evaluate the derivative of a composite function.


In calculus, the chain rule and product rule are both essential parts of derivatives. According to the above discussion, we can conclude that the chain rule can apply to composite and the product of two functions. However, we cannot use the product rule for composite parts as it is only reliable for the product of two or more functions.

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