1. Differentiation
Product Rule
By Mr Porter

2. Important Information.
To help explain the Product Rule used in differentiation, I will be using the following derivatives.
Exponential Function:
Trigonometic Sine Function:

3. Definition of the Product Rule.
If u = f(x) and v = g(x), then
Using the dash notation: If y = uv, then
Example: Use the Product Rule to differentiate
The easy way to do this problem would be to simplify the algebra before differentiating.

4. Example 1 Differentiate the following, given that Solution Solution
Let u = x3 and v = ex
Let u = 4x -2 and v = ex
then u’ = 3x2 and v’ = ex
then u’ = -8x -3 and v’ = ex
y’ = uv’ + vu’
y’ = uv’ + vu’
y’ = 4x -2 (ex) +ex (-8x -3)
y’ = x3 (ex) +ex (3x2)
y’ = x3 ex + 3x2ex
factorise
y’ = 4x -2 ex – 8x -3ex
factorise
y’ = x2 ex (x + 3)
y’ = 4x -2 ex (1 – 2x -1)

5. Example 2 Differentiate the following, given that Solution
Let u = 4x 3 and v = sin x
Solution
Let u = -5x and v = sin x
then u’ = -5 and v’ = cos x
y’ = uv’ + vu’
then u’ = 12x 2 and v’ = cos x
y’ = uv’ + vu’
y’ = -5x (cos x) + (-5) sin x
y’ = 4x 3 (cos x) + (12x 2) sin x
y’ = -5x cos x – 5sin x
factorise
y’ = 4x 3 cos x + 12x2 sin x
factorise
y’ = -5(x cos x + sin x)
y’ = 4x 2(x cos x + 3 sin x)

6. Example 3: Combination of Product Rule and Chain Rule.
Chain Rule is the Function of a Function Rule.
Differentiate y = x2 (3x + 1)4
Now y’ = uv’ + vu’
y’ = x2 x 12(3x + 1)3 +2x (3x + 1)4
Solution
Let u = x2 and v = (3x + 1)4
y’ = 12x2(3x + 1)3 + 2x (3x + 1)4
factorise
y’ = 2x(3x + 1)3[6x + (3x + 1)]
y’ = 2x(3x + 1)3[6x + 3x + 1]
Differentiate the brackets, multiply by the derivative of what’s inside!
y’ = 2x(3x + 1)3(9x + 1)

7. Example 4: Combination of Product Rule and Chain Rule.
Differentiate y = -2x4 (5 – x2)3
Solution
Let u = -2x4 and v = (5 - x2)3
Now y’ = uv’ + vu’
y’ = -2x4 x -6x(5 – x2)2 + -8x3 (5 - x2)3
y’ = 12x5(5 – x2)2 – 8x3(5 - x2)3
factorise
Differentiate the brackets, multiply by the derivative of what’s inside!
y’ = 4x3(5 - x2)2[3x2 – 2(5 – x2)]
y’ = 4x3(5 - x2)2 [3x2 – x2]
y’ = 4x3(5 - x2)2 (5x2 – 10)
factorise
y’ = 20x3(5 - x2)2 (x2 – 2)