## Product Rule

### The product rule is a fundamental rule in calculus that allows us to find the derivative of a product of two functions

The product rule is a fundamental rule in calculus that allows us to find the derivative of a product of two functions. Let’s consider two functions, f(x) and g(x), and their product h(x) = f(x) * g(x).

The product rule states that the derivative of the product h(x) can be found by taking the derivative of the first function, f(x), multiplied by the second function, g(x), and then adding to it the derivative of the second function, g(x), multiplied by the first function, f(x). Mathematically, it can be expressed as:

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

In other words, to find the derivative of a product, we differentiate each function separately and then combine the results using multiplication and addition. This rule is especially useful when dealing with functions that involve multiplication, such as polynomial functions or exponential functions.

Let’s illustrate the product rule with an example. Consider the functions f(x) = x^2 and g(x) = sin(x). We want to find the derivative of their product h(x) = x^2 * sin(x).

First, we differentiate the first function f(x) = x^2, which gives us f'(x) = 2x.

Next, we differentiate the second function g(x) = sin(x), which gives us g'(x) = cos(x).

Finally, we apply the product rule formula to find the derivative of h(x):

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

= (2x * sin(x)) + (x^2 * cos(x))

So, the derivative of h(x) = x^2 * sin(x) is h'(x) = (2x * sin(x)) + (x^2 * cos(x)).

The product rule allows us to differentiate a wide range of functions that involve multiplication, making it an essential tool in calculus.

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