Using the product rule to find the derivative of two functions | A comprehensive guide and example.

Product Rule

The product rule is a formula used to find the derivative of a product of two functions

The product rule is a formula used to find the derivative of a product of two functions. In calculus, the derivative of a function gives us the rate at which the function is changing at any point.

The product rule is stated as follows: If we have two functions, u(x) and v(x), both of which are differentiable with respect to x, then the derivative of their product, u(x) * v(x), is given by the formula:

(d/dx)(u(x) * v(x)) = u'(x) * v(x) + u(x) * v'(x)

In other words, to find the derivative of a product of two functions, we take the derivative of the first function (u'(x)), multiply it by the second function (v(x)), and add it to the product of the first function (u(x)) multiplied by the derivative of the second function (v'(x)).

The product rule is very useful in calculus, as it allows us to find the derivatives of functions involving multiplication easily. It is often used when finding the derivatives of expressions such as polynomials, trigonometric functions, and exponential functions.

Here’s an example to illustrate the use of the product rule:

Let’s say we have the functions u(x) = 2x^2 and v(x) = sin(x). We want to find the derivative of their product, u(x) * v(x).

Using the product rule, we have:

(d/dx)(u(x) * v(x)) = u'(x) * v(x) + u(x) * v'(x)

Taking derivatives, u'(x) = 4x and v'(x) = cos(x). Plugging these values in, we get:

(d/dx)(u(x) * v(x)) = (4x) * sin(x) + (2x^2) * cos(x)

Simplifying further, we have:

(d/dx)(u(x) * v(x)) = 4x * sin(x) + 2x^2 * cos(x)

This is the derivative of the product of the functions u(x) = 2x^2 and v(x) = sin(x).

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