What is product rule in differentiation?

Differentiation is the mathematical process of finding the rate at which a function changes with respect to its independent variable. The product rule allows us to directly differentiate the product of two functions without expanding or simplifying the expression. It saves time and effort by providing a systematic approach to computing the derivative of a product.

You can see here for more details of other differentiation rules.

Product rule mathematically

The product rule is a fundamental rule in Calculus used to differentiate the product of two functions. It states that the derivative of the product of two functions is equal to the derivative of the first function times the second function plus the first function times the derivative of the second function. Mathematically, if we have two functions, $f(x)$ and $g(x)$, the product rule can be expressed as:

Examples of the product rule

The product rule is needed in certain situations where further simplifying the expression is impossible. It allows us to understand why the rule is important and how it helps us handle complexities in those specific cases.

Example 1: Exponential function

Let's consider a function $h(x)$ that cannot be simplified any further:

Solution: Let's call $f(x) = x$ and $g(x) = e^{x}$

Example 2: Trigonometric functions

This example demonstrates how powerful the product rule is for trigonometric functions where further simplification isn't possible.

Solution: Again, using a similar approach, we can say $f(x) = sin(x)$ and $g(x) = cos(x)$

Quiz

Now that you know the product rule in differentiation, you can challenge yourself with a quiz.

Conclusion

The product rule simplifies the process of finding derivatives. It enables us to differentiate a wide range of functions efficiently, making it an essential tool in Calculus and mathematical analysis.