Derivative of a Product

How to find the derivative of a product f(x)g(x): formula (product rule), 1 example, and its solution.

Formula

Derivative of f(x)g(x): Product Rule

The derivative of f(x)g(x) is f'(x)g(x) + f(x)g'(x).

First differentiate the former term: f'(x)g(x).
Then differentiate the latter term: +f(x)g'(x).

Example

Example

Solution

The given function is the product of (2x3 - 5) and (4x2 + x).

Then y' is equal to,
write the derivative of (2x3 - 5), 2⋅3x2 - 0.

Derivative of a Polynomial

Write (4x2 + x).

Write +.

Write (2x3 - 5).

Write, the derivative of (4x2 + x), 4⋅2x1 + 1.

So y' = (2⋅3x2 - 0)(4x2 + x) + (2x3 - 5)(4⋅2x1 + 1).

(2⋅3x2 - 0) = 6x2
(4⋅2x1 + 1) = (8x + 1)

6x2(4x2 + x) = 24x4 + 6x3

+(2x3 - 5)(8x + 1) = +16x4 + 2x3 - 40x - 5

FOIL Method

24x4 + 16x4 = 40x4
+6x3 + 2x3 = +8x3

So y' = 40x4 + 8x3 - 40x - 5 is the derivative of the function.