Mastering the Product Rule: How to Differentiate the Product of Two Functions in Calculus

product rule

The product rule is a method used in calculus to differentiate the product of two functions

The product rule is a method used in calculus to differentiate the product of two functions.

Let’s say we have two functions, f(x) and g(x), and we want to find the derivative of their product, h(x) = f(x) * g(x).

The product rule states that the derivative of the product of two functions is equal to the derivative of the first function multiplied by the second function, plus the first function multiplied by the derivative of the second function. Mathematically, the product rule can be expressed as:

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

To better understand this, let’s walk through an example.

Example: Find the derivative of h(x) = x^2 * sin(x).

In this case, f(x) = x^2 and g(x) = sin(x).

First, let’s find the derivatives of each function:
f'(x) = 2x (using the power rule for derivatives)
g'(x) = cos(x) (using the derivative of sin(x) = cos(x))

Now, apply the product rule:

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

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

The product rule can be very useful when dealing with functions that are products of other functions, as it allows us to find their derivatives without having to expand the product and apply derivative rules separately.

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