What is product rule?

The product rule is a fundamental concept in calculus used to find the derivative of a product of two or more functions.

Formula:

If you have two functions, u(x) and v(x), the derivative of their product, y = u(x)v(x), is given by:

dy/dx = u'(x)v(x) + u(x)v'(x)

Where u'(x) and v'(x) represent the derivatives of u(x) and v(x) respectively.

In simpler terms:

(First function derivative * Second function) + (First function * Second function derivative)

Generalization:

The product rule can be extended to more than two functions. For example, if y = u(x)v(x)w(x), then:

dy/dx = u'(x)v(x)w(x) + u(x)v'(x)w(x) + u(x)v(x)w'(x)

When to Use:

Use the product rule whenever you need to find the derivative of a function that is expressed as the product of two or more distinct functions. For example, when differentiating x*sin(x) or (x^2 + 1)*e^x.

Example:

Let y = x^2 * sin(x)

• u(x) = x^2
• v(x) = sin(x)

Then:

• u'(x) = 2x
• v'(x) = cos(x)

Applying the product rule:

dy/dx = (2x * sin(x)) + (x^2 * cos(x))

Importance:

The product rule is essential for differentiating a wide range of functions and is a building block for more advanced differentiation techniques like the quotient%20rule and chain%20rule.