Step by Step
1
The rule itself
For f(x)·g(x), the derivative is f·g' + g·f' — first function times the derivative of the second, plus the second function times the derivative of the first.
2
Why you can't just multiply the derivatives
A common instinct is to just differentiate each piece and multiply the results — this is always wrong. The product rule exists precisely because the derivative of a product isn't the product of the derivatives.
3
Identify "first" and "second" clearly
Before applying the rule, clearly label which function is f (first) and which is g (second) — this keeps you from mixing up terms partway through.
4
Simplify after applying the rule
After applying the product rule, you'll often be able to factor or combine terms — don't stop at the raw formula output if the expression can be simplified further.
Applied Walkthrough
1
Differentiate h(x) = x²·sin(x). Let f = x² (so f' = 2x) and g = sin(x) (so g' = cos(x)).
2
Apply the product rule: h'(x) = f·g' + g·f' = x²·cos(x) + sin(x)·2x.
3
Simplify slightly: h'(x) = x²cos(x) + 2x·sin(x).
4
Contrast this with the wrong approach: simply multiplying 2x (the derivative of x²) by cos(x) (the derivative of sin(x)) would give 2x·cos(x) — a completely different and incorrect answer.
Exam Application
Exams test whether you can correctly apply the full product rule formula rather than simply multiplying the two derivatives together, and whether you can identify which function is "first" and "second" without mixing them up.
⚠ Common Trap
The single biggest trap: multiplying the two derivatives together (f'·g') instead of using the full formula f·g' + g·f'. This shortcut is never correct for a product of two functions.
✓ Quick Self-Check
1. What is the product rule formula?
d/dx[f·g] = f·g' + g·f'.
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2. Differentiate h(x) = x²·sin(x).
h'(x) = x²cos(x) + 2x·sin(x).
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3. Is it ever correct to just multiply f'(x) by g'(x) for a product?
No — that always gives the wrong answer; the full product rule formula must be used.
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4. In the product rule, what does f·g' represent?
The first function times the derivative of the second function.
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5. What should you do after applying the product rule formula?
Simplify or factor the resulting expression where possible.
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