Derivative of a Product
How to find the derivative of a product f(x)g(x): formula (product rule), 1 example, and its solution.
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 = (2x3 - 5)(4x2 + x), y' = ?
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 (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
24x4 + 16x4 = 40x4
+6x3 + 2x3 = +8x3
y' = 40x4 + 8x3 - 40x - 5
is the derivative of the function.