The Product Rule (VCE SSCE Mathematical Methods): Revision Notes

Worked Example: Simple Product

Consider $F(x) = (x^2 + 3x)(4x + 5)$. This is a product of two functions:

• $f(x) = x^2 + 3x$
• $g(x) = 4x + 5$

Using the product rule:

Note: This could also be found by expanding the brackets first and then differentiating, but the product rule becomes essential when dealing with more complex functions where expansion isn't practical.

Worked Example: Differentiating Polynomial Products

Differentiate each of the following with respect to $x$:

a) $(2x^2 + 1)(5x^3 + 16)$

b) $x^3(3x - 5)^4$

Solution

a) Let $y = (2x^2 + 1)(5x^3 + 16)$

Set $u = 2x^2 + 1$ and $v = 5x^3 + 16$

Then $\frac{du}{dx} = 4x$ and $\frac{dv}{dx} = 15x^2$

Using the product rule:

b) Let $y = x^3(3x - 5)^4$

Set $u = x^3$ and $v = (3x - 5)^4$

Then $\frac{du}{dx} = 3x^2$ and $\frac{dv}{dx} = 12(3x - 5)^3$ (using the chain rule)

Using the product rule:

Worked Example: Products with Negative Powers

For $F: \mathbb{R} \setminus \{0\} \to \mathbb{R}$, $F(x) = x^{-3}(10x^2 - 5)^3$, find $F'(x)$.

Solution

Let $f(x) = x^{-3}$ and $g(x) = (10x^2 - 5)^3$

Then $f'(x) = -3x^{-4}$ and $g'(x) = 60x(10x^2 - 5)^2$ (using the chain rule)

By the product rule:

Worked Example: Products with Exponential Functions

Differentiate each of the following with respect to $x$:

a) $e^x(2x^2 + 1)$

b) $e^x\sqrt{x - 1}$

Solution

a) Let $y = e^x(2x^2 + 1)$

Using the product rule:

b) Let $y = e^x\sqrt{x - 1}$

Using the product rule and chain rule:

Worked Example: Products with Trigonometric Functions

Find the derivative of each of the following with respect to $x$:

a) $2x^2\sin(2x)$

b) $e^{2x}\sin(2x + 1)$

c) $\cos(4x)\sin(2x)$

Solution

a) Let $y = 2x^2\sin(2x)$

Applying the product rule:

b) Let $y = e^{2x}\sin(2x + 1)$

Applying the product rule:

c) Let $y = \cos(4x)\sin(2x)$

The product rule gives: