The intuition behind f′g + fg′

• 💡 Picture a rectangle with sides f(x) and g(x) — its area is f·g.
• 🎯 If both sides are changing, the area changes in two ways: because f is growing (f′·g) AND because g is growing (f·g′).
• ✅ Add both effects together — that’s exactly the Product Rule formula.

One memorable line: the rectangle grows from both edges at once, so you add both contributions.

Where students lose points

• ❌ Multiplying f′(x) · g′(x) — this is NOT the derivative of a product.
• ❌ Forgetting to keep the original (non-derivative) function in each term.
• ❌ Dropping a term entirely when g(x) is a constant (remember: derivative of a constant is 0, so that whole term vanishes — but don’t forget the other one).
• ✅ Always write both terms first, then simplify.

Differentiate y = x² · sin(x)

• Let f(x) = x² → f′(x) = 2x
• Let g(x) = sin(x) → g′(x) = cos(x)
• Apply the rule: y′ = f′g + fg′
• y′ = (2x)(sin x) + (x²)(cos x)

✅ Final answer: y′ = 2x·sin(x) + x²·cos(x)

Two different jobs

• 🎯 Product Rule: use when two separate functions are multiplied — f(x)·g(x).
• 🎯 Chain Rule: use when one function is nested inside another — f(g(x)).
• 💡 Quick test: can you rewrite the expression as ‘this times that’? Product Rule. Is it ‘this, plugged into that’? Chain Rule.
• ✅ Many real problems need both rules together — that’s normal, not a sign you’re doing something wrong.

Copy-paste recipe

• 1️⃣ Identify f(x) and g(x) — the two pieces being multiplied.
• 2️⃣ Find f′(x) and g′(x) separately.
• 3️⃣ Plug into: f′(x)·g(x) + f(x)·g′(x)
• 4️⃣ Simplify if possible.

💡 Keep this four-step recipe taped above your desk until it’s automatic.