Differentiation Product Rule- Step-by-Step Examples and Practice

What the Product Rule Actually Is

The product rule is a differentiation technique for functions that multiply together. If you have two functions multiplied, you can't just differentiate each and multiply the results.

The formula:

d/dx [f(x) · g(x)] = f'(x) · g(x) + f(x) · g'(x)

Or in Leibniz notation: (fg)' = f'g + fg'

That's it. One formula. The hard part is applying it correctly.

Why You Can't Just Wing It

Most students see "multiply" and assume they can just multiply derivatives. They can't.

Example of a wrong approach:

Given: x² · sin(x)

Wrong: differentiate each, multiply = 2x · cos(x)

Right: apply the product rule = 2x · sin(x) + x² · cos(x)

These are not the same thing. The wrong answer will cost you points.

Step-by-Step Examples

Example 1: Basic Polynomial × Trig

Find the derivative of f(x) = x³ · cos(x)

Step 1: Identify your two functions

• f(x) = x³
• g(x) = cos(x)

Step 2: Differentiate each

• f'(x) = 3x²
• g'(x) = -sin(x)

Step 3: Plug into the formula

f'(x)g(x) + f(x)g'(x)

= 3x² · cos(x) + x³ · (-sin(x))

= 3x²cos(x) - x³sin(x)

Example 2: Exponential × Logarithm

Find the derivative of f(x) = eˣ · ln(x)

Step 1: Two functions

• f(x) = eˣ
• g(x) = ln(x)

Step 2: Differentiate

• f'(x) = eˣ
• g'(x) = 1/x

Step 3: Apply the rule

= eˣ · ln(x) + eˣ · (1/x)

= eˣln(x) + eˣ/x

= eˣ(ln(x) + 1/x)

Example 3: Three Functions Multiplied

When you have three functions, pick two, apply the rule, then apply it again.

Find the derivative of f(x) = x² · sin(x) · eˣ

First: Treat x² · sin(x) as one function (call it u)

u = x² · sin(x), so u' = 2x·sin(x) + x²·cos(x)

Then: Apply product rule to u · eˣ

= u' · eˣ + u · eˣ

= [2x·sin(x) + x²·cos(x)] · eˣ + x²·sin(x) · eˣ

= eˣ[2x·sin(x) + x²·cos(x) + x²·sin(x)]

The "FOIL" Confusion

Students often mix up product rule with FOIL. They are not the same thing.

FOIL gives you an expression. Product rule gives you a derivative. Different operations, different tools.

Common Mistakes That Will Kill Your Answer

• Forgetting one of the two terms — you need BOTH f'g AND fg'
• Multiplying derivatives instead of adding — product rule is addition, not multiplication
• Wrong derivative of one function — check your trig, log, and power rules before applying product rule
• Not simplifying at the end — factoring out common terms makes your answer cleaner and shows you understand what you're doing

Product Rule vs. Other Rules

Getting Started: How to Apply the Product Rule

Step 1: Look at the function. Identify the two main parts being multiplied. Label one "f" and one "g".

Step 2: Differentiate f to get f'. Differentiate g to get g'. Don't rush this step. If your basic derivatives are wrong, everything falls apart.

Step 3: Write out f'g + fg'. This is your unsimplified answer.

Step 4: Simplify if possible. Factor out common terms. Combine like terms. A simplified answer is easier to check and easier to grade.

Step 5: Verify. One quick check: if x is in your original function, try plugging in a simple value like x=1. If your derivative is correct, the slope estimate should be reasonable.

Practice Problems

Try these before checking the answers:

1. f(x) = (3x + 1)(x² - 2)
2. f(x) = x⁴ · eˣ
3. f(x) = sin(x) · cos(x)
4. f(x) = √x · ln(x)

Answers:

1. 9x² + 2x - 6 (you can also expand first, then differentiate)
2. x³eˣ(4 + x) or 4x³eˣ + x⁴eˣ
3. cos²(x) - sin²(x) (which equals cos(2x))
4. (1/2√x)ln(x) + √x · (1/x) = ln(x)/(2√x) + 1/√x

The Bottom Line

Product rule isn't complicated. It's mechanical. Identify your functions, differentiate them separately, then add the two resulting terms.

The mistakes come from rushing the identification step or making errors in basic derivatives. Get those right and you'll get full credit.