The Product Rule: Differentiation Made Easy

📜 Formula of the Product Rule

If a function $f(x)$ is the product of two functions $u(x)$ and $v(x)$, the formula is:

💡Good to know: The order of $u$ and $v$ does not matter, as multiplication is commutative. However, both terms must always be included — differentiating only one part is not enough!

🪜 Application in 3 Simple Steps

You can always apply the product rule when two functions are multiplied by each other.

1. Identify the factors: Break the function down into its two factors $u(x)$ and $v(x)$.
2. Differentiate each factor: Compute $u'(x)$ and $v'(x)$ using the known differentiation rules (such as the power rule or trigonometric derivatives).
3. Apply the product rule formula: Insert the results into $f'(x) = u'(x)\cdot v(x) + u(x)\cdot v'(x)$.

✏️ Examples for Illustration

The product rule is best understood with practical examples.

Example 1: Polynomial × Polynomial

Given the function: $f(x) = (x^2 + 1)(x^3 - 2)$

• 1. Identify: $u(x) = x^2 + 1$, $v(x) = x^3 - 2$.
• 2. Differentiate: $u'(x) = 2x$, $v'(x) = 3x^2$.
• 3. Apply the formula: $$ f'(x) = u'(x)v(x) + u(x)v'(x) $$ $$ = (2x)(x^3 - 2) + (x^2 + 1)(3x^2) $$ $$ = 2x^4 - 4x + 3x^4 + 3x^2 $$ $$ = 5x^4 + 3x^2 - 4x $$

So the result is: $f'(x) = 5x^4 + 3x^2 - 4x$

Example 2: Polynomial × Trigonometric Function

Given the function: $f(x) = x^2 \sin(x)$

• 1. Identify: $u(x) = x^2$, $v(x) = \sin(x)$.
• 2. Differentiate: $u'(x) = 2x$, $v'(x) = \cos(x)$.
• 3. Apply the formula: $$ f'(x) = (2x)\sin(x) + (x^2)\cos(x) $$

So the result is: $f'(x) = 2x\sin(x) + x^2\cos(x)$

Example 3: Exponential × Polynomial

Given the function: $f(x) = e^x \cdot x^3$

• 1. Identify: $u(x) = e^x$, $v(x) = x^3$.
• 2. Differentiate: $u'(x) = e^x$, $v'(x) = 3x^2$.
• 3. Apply the formula: $$ f'(x) = e^x \cdot x^3 + e^x \cdot 3x^2 $$ $$ = e^x (x^3 + 3x^2) $$

So the result is: $f'(x) = e^x (x^3 + 3x^2)$