Mastering the Product Rule in Calculus | A Comprehensive Guide to Differentiating the Product of Functions

Product Rule

The product rule is a formula used in calculus to differentiate the product of two or more functions

The product rule is a formula used in calculus to differentiate the product of two or more functions. It allows us to find the derivative of the product of two functions without having to explicitly multiply them. The product rule is often used when you have a function that can be expressed as the product of two simpler functions.

Mathematically, if we have two functions f(x) and g(x), the product rule states that the derivative of their product, denoted as (f(x) * g(x))’, is given by:

(f(x) * g(x))’ = f'(x) * g(x) + f(x) * g'(x)

Here, f'(x) denotes the derivative of the function f(x) with respect to x, and g'(x) denotes the derivative of the function g(x) with respect to x.

To apply the product rule, you would differentiate each individual function, multiply them, and then add them together as shown in the formula. This allows us to find the derivative of the product function without explicitly multiplying the two functions.

For example, let’s say we have the function h(x) = x^2 * sin(x). To find the derivative of this function using the product rule, we would first differentiate x^2 to get 2x, and then differentiate sin(x) to get cos(x). Finally, we can apply the product rule:

h'(x) = (x^2 * sin(x))’
= (2x * sin(x)) + (x^2 * cos(x))
= 2x * sin(x) + x^2 * cos(x)

Therefore, the derivative of h(x) is 2x * sin(x) + x^2 * cos(x) according to the product rule.

In summary, the product rule is a powerful tool in calculus that allows us to differentiate the product of two or more functions by differentiating each function and then applying a specific formula.

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