Question

How is the chain rule different from the product rule?

Answer 1

Here we will write the definition of the chain rule and the product rule with example. Then by using these definitions, we will get the difference between the chain rule and product rule. Both rules are used in the method of differentiation. In order to find derivatives of a function, we generally use the product rule first and then the chain rule.

Complete step by step solution:
We will write the basic definitions of the chain rule and product rule. Therefore, we get
Chain rule is a rule which is used to differentiate the composition of the functions i.e. when the function is to be differentiated is in the form of $F\left( {f\left( x \right)} \right)$.
For example: let $F\left( x \right) = x + 1$ and $f\left( x \right) = 2x + 2$. So the composition of the function is $F\left( {f\left( x \right)} \right) = F\left( {2x + 2} \right) = \left( {2x + 2} \right) + 1 = 2x + 3$.
The product rule is a rule which is used to differentiate the product of the two functions i.e. when the function is to be differentiated is in the form of $f\left( x \right) \cdot g\left( x \right)$.
For example: let $f\left( x \right) = x + 1$ and $g\left( x \right) = 2x + 2$. So the composition of the function is $f\left( x \right) \cdot g\left( x \right) = \left( {x + 1} \right) \cdot \left( {2x + 2} \right)$.
Hence the main difference between the chain rule and product rule is that it is used for the differentiation of the function of a function and the other is used for the differentiation of the product of the function.

Note:
We should remember that a differentiable function is always continuous but the converse is not true which means a function may be continuous but not always differentiable. A differentiable function may be defined as is a function whose derivative exists at every point in its range of domain. Differentiation is the opposite of integration i.e. differentiation of the integration is equal to the value of the function or vice versa.