# Derivative of a Product

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

## FormulaProduct 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).

## Exampley = (2x^{3} - 5)(4x^{2} + x), y' = ?

The given function is the product of (2x^{3} - 5) and (4x^{2} + x).

Then y' is equal to,

write the derivative of (2x^{3} - 5), 2⋅3x^{2} - 0.

Derivative of a Polynomial

Write (4x^{2} + x).

Write +.

Write (2x^{3} - 5).

Write, the derivative of (4x^{2} + x), 4⋅2x^{1} + 1.

So y' = (2⋅3x^{2} - 0)(4x^{2} + x) + (2x^{3} - 5)(4⋅2x^{1} + 1).

(2⋅3x^{2} - 0) = 6x^{2}

(4⋅2x^{1} + 1) = (8x + 1)

6x^{2}(4x^{2} + x) = 24x^{4} + 6x^{3}

+(2x^{3} - 5)(8x + 1) = +16x^{4} + 2x^{3} - 40x - 5

FOIL Method

24x^{4} + 16x^{4} = 40x^{4}

+6x^{3} + 2x^{3} = +8x^{3}

So

y' = 40x^{4} + 8x^{3} - 40x - 5

is the derivative of the function.