Mastering the Product Rule in Calculus: A Guide to Differentiating the Product of Two Functions

Product Rule

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

The product rule is a fundamental rule of differentiation in calculus that allows us to find the derivative of a product of two functions. Let’s say we have two functions, u(x) and v(x), and we want to find the derivative of their product, which is represented by w(x) = u(x) * v(x).

The product rule states that the derivative of w(x) is given by the following formula:

w'(x) = u(x) * v'(x) + v(x) * u'(x)

This formula shows that the derivative of the product is equal to the first function times the derivative of the second function, plus the second function times the derivative of the first function.

To use the product rule, you need to identify u(x) and v(x), and then find their derivatives u'(x) and v'(x). Once you have these values, you can substitute them into the formula and simplify to find the derivative of the product.

Let’s work through an example to illustrate how to use the product rule:

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

In this case, u(x) = x^2 and v(x) = sin(x).

Step 1: Find the derivatives of u(x) and v(x).
u'(x) = 2x
v'(x) = cos(x)

Step 2: Substitute these values into the product rule formula.
f'(x) = x^2 * cos(x) + sin(x) * 2x

Step 3: Simplify the expression if necessary.
f'(x) = 2x * sin(x) + x^2 * cos(x)

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

That’s the basic concept of the product rule. It allows us to differentiate products of functions and is an essential tool in calculus. Remember to always identify the two functions and their derivatives correctly, then apply the formula to find the derivative of the product.

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