Product Rule

📖 What is the Product Rule?

The product rule is used to differentiate a product of two functions. You cannot simply multiply the two individual derivatives — you must apply this rule.

            (f · g)' = f · g' + g · f'
          

In words: "first times derivative of second, plus second times derivative of first."

🔤 What Each Symbol Means

f

First functionAny differentiable function of x

g

Second functionAny differentiable function of x

f'

Derivative of fdf/dx — how f changes with x

g'

Derivative of gdg/dx — how g changes with x

📝 Step-by-Step Example 1

Differentiate: y = x² · sin x

1

Identify f and gf = x² g = sin x

2

Find the derivatives f' and g'f' = 2x g' = cos x

3

Apply the product rule: fg' + gf'dy/dx = x²·cos x + sin x·2x

✓

Answer: dy/dx = x²cos x + 2x sin x

📝 Step-by-Step Example 2

Differentiate: y = eˣ · ln x

1

Identify f and gf = eˣ g = ln x

2

Find the derivativesf' = eˣ g' = 1/x

3

Apply the product ruledy/dx = eˣ·(1/x) + ln x·eˣ

✓

Answer: dy/dx = eˣ(1/x + ln x)

⚠️ Common Mistake

Wrong:

d/dx(fg) = f' · g' ✗
You cannot multiply the derivatives directly.

Correct:

d/dx(fg) = f·g' + g·f' ✓

💡 Extended Product Rule (Three Functions)

For three functions multiplied together:

(fgh)' = f'gh + fg'h + fgh'

Each term has one function differentiated and the rest left unchanged.