Product and quotient rules: one example worked line by line for each, and the sign error in the quotient rule that costs the mark

The product rule says (fg)' = f'g + fg': differentiate each factor in turn and add. The quotient rule says (f/g)' = (f'g − fg')/g²: the same two pieces, subtracted in that order, over the square of the bottom. The order matters only in the quotient rule, and reversing it is the single most common lost mark in a derivatives test. Here is one example of each, worked line by line, with the wrong line named.

The product rule and the quotient rule formula, and how to check you have them the right way round

For two functions f and g of x, with f' and g' their derivatives:

• Product rule: (fg)' = f'g + fg'
• Quotient rule: (f/g)' = (f'g − fg')/g²

The product rule is safe either way round, because its two pieces are added. The quotient rule is not, and here is a ten-second check that tells you whether you have the order right. Differentiate 1/x two ways.

As a power, 1/x = x⁻¹, and the power rule gives −1 · x⁻² = −1/x². Now with the quotient rule, f = 1 and g = x, so f' = 0 and g' = 1: (0 · x − 1 · 1)/x² = −1/x². The two agree. If your version of the quotient rule gives +1/x², the subtraction is reversed. Run this check before every quotient-rule question until the order is automatic; it costs nothing and it is the page's first offer.

The proofs of both rules come from the limit definition of the derivative and are in the textbook chapter; you will not be asked to reproduce them in MCV4U, and this page does not.

Why there is a product rule at all

A new student's first instinct is that (fg)' should be f'g'. One line shows why it is not. Take f = x and g = x. Then f'g' = 1 · 1 = 1. But fg = x², whose derivative is 2x. One is not 2x, so multiplying the derivatives is wrong.

The reason is that both factors change at once. When x moves a little, f changes and g changes, and the product changes by f's change scaled by g's current value plus g's change scaled by f's current value. That is exactly f'g + fg'. The rule is not a convention; it is what happens.

Worked example 1: the product rule, line by line

Find h'(x) for h(x) = (3x + 1)(x² − 4).

• Line 1, name the factors. f = 3x + 1, g = x² − 4.
• Line 2, differentiate each. f' = 3, g' = 2x.
• Line 3, write the rule. f'g + fg' = 3(x² − 4) + (3x + 1)(2x).
• Line 4, expand. 3x² − 12 + 6x² + 2x.
• Line 5, collect. h'(x) = 9x² + 2x − 12.

Now the check, and this is a habit worth teaching yourself: expand first and differentiate without the rule. h(x) = (3x + 1)(x² − 4) = 3x³ − 12x + x² − 4 = 3x³ + x² − 12x − 4. The power rule gives 9x² + 2x − 12. Same answer. Whenever both factors are polynomials you can check a product-rule answer this way, and you should, until the rule is reliable.

Worked example 2: the quotient rule, line by line

Find q'(x) for q(x) = (x² + 1)/(2x − 3).

• Line 1, name the top and the bottom. f = x² + 1 (top), g = 2x − 3 (bottom).
• Line 2, differentiate each. f' = 2x, g' = 2.
• Line 3, write the numerator in the right order. f'g − fg' = 2x(2x − 3) − (x² + 1)(2).
• Line 4, expand the numerator. 4x² − 6x − 2x² − 2 = 2x² − 6x − 2.
• Line 5, put it over the bottom squared. q'(x) = (2x² − 6x − 2)/(2x − 3)².

Leave the denominator as (2x − 3)², squared and unexpanded. Expanding it gains nothing and hides the structure a marker is looking for.

The common wrong lines

The product rule as f'g'. On example 1 this gives 3 · 2x = 6x, which is not 9x² + 2x − 12. The x · x test above is the whole explanation: multiplying the derivatives is not the derivative of the product, because both factors are changing at once.

The quotient rule's subtraction reversed. Writing fg' − f'g instead of f'g − fg' gives (x² + 1)(2) − 2x(2x − 3) = 2x² + 2 − 4x² + 6x = −2x² + 6x + 2, so the answer comes out as (−2x² + 6x + 2)/(2x − 3)², the exact negative of the truth. Every sign is wrong and the algebra looks fine, which is why this error survives to the exam. Run the 1/x check: the reversed order gives +1/x², and 1/x is a decreasing function for x > 0, so its derivative must be negative. The order that makes 1/x come out negative is the right one.

The denominator not squared, or the numerator squared too. The quotient rule squares the bottom and only the bottom. A numerator that has been squared, or a bottom left as 2x − 3, is a lost mark even when the subtraction order is right.

Booth, Lange, Koedinger and Newton (2013) studied algebra students who worked through incorrect examples and explained the fault, alone or alongside correct examples, and found that the incorrect examples were especially good for the students' understanding of why a procedure works. That is why this page shows the reversed order and says why it fails, rather than only showing the right order. Reading a wrong line with its reason attached makes it easier to notice in your own work.

Siblings to try cold

Cover the two examples. Do both of these without looking back, naming f and g on the first line each time, and run the 1/x check in your head before you write the quotient's numerator. Test yourself before you look is the rule that makes the answers below worth having.

• Product: p(x) = (2x − 5)(x² + 3x).
• Quotient: r(x) = (x³ − 1)/(x + 2).

Check

• p'(x) = 6x² + 2x − 15. Check by expanding: p(x) = 2x³ + 6x² − 5x² − 15x = 2x³ + x² − 15x, whose derivative is 6x² + 2x − 15.
• r'(x) = (2x³ + 6x² + 1)/(x + 2)². The numerator is 3x²(x + 2) − (x³ − 1) = 3x³ + 6x² − x³ + 1 = 2x³ + 6x² + 1.

If your quotient numerator came out as −2x³ − 6x² − 1, the order was reversed. If it came out as 2x³ + 6x² − 1, the double negative on the −1 was missed: minus (x³ − 1) is −x³ + 1.

When to use the quotient rule, and when not to

The quotient rule is for a changing top over a changing bottom that will not split apart. In three common cases it is the long way round.

• A constant on top. 5/x² is 5x⁻², and the power rule gives −10x⁻³ = −10/x³ in one line. No subtraction order to get wrong.
• A single term on the bottom. (x² + 3)/x splits into x + 3/x = x + 3x⁻¹, and each term is a power rule: 1 − 3/x².
• A function to a negative power. 1/(x² + 1) is (x² + 1)⁻¹, which is the chain rule: −1 · (x² + 1)⁻² · 2x = −2x/(x² + 1)².

If none of the three applies, the quotient rule it is. Its first real job in MCV4U comes at idea 11, where tan x = sin x/cos x gives (cos x · cos x − sin x · (−sin x))/cos²x = (cos²x + sin²x)/cos²x = 1/cos²x = sec²x. That is where the sec²x in the complete list of derivative rules comes from.

Where this sits in the course, and where it goes next

In MCV4U this is idea 7 of 24, Product and quotient rules. Before it is idea 6, Power rule; derivatives of polynomials, which every line above leans on. After it is idea 8, Chain rule, which is what the product rule becomes when the second factor sits inside the first instead of beside it. The MCV4U course map shows all 24 ideas in teaching order.

In a first-year Calculus I course the same material is idea 5 of 26, Power, product and quotient rules, between idea 4, The derivative as a limit, and idea 6, The chain rule. First-year calculus, what changes is written for the Grade 12 student about to meet it.

Both examples on this page are polynomials on purpose. MCV4U meets the product and quotient rules before the trig and exponential derivatives, so nothing here needs a function you have not seen. When those arrive, the rules are the same; only f and g change.

Practise the two rules mixed with the power rule and the chain rule, not in a block of ten quotients. Rohrer and Taylor (2007) found that students who practised shuffled problem types scored lower during practice and higher on the test than students who practised in blocks, because a block tells you the rule before you read the problem. Practise the rules mixed has the method, and worked examples, then practice problems has the order; Sweller and Cooper (1985) is the reason the examples here come before the siblings.

Product and quotient rules on one screen

• Product: (fg)' = f'g + fg'. Order does not matter; the pieces are added.
• Quotient: (f/g)' = (f'g − fg')/g². Order matters; the bottom alone is squared.
• The 1/x check: the quotient rule on 1/x must give −1/x². If you get +1/x², the subtraction is reversed.
• Why not f'g': x · x = x² has derivative 2x, not 1 · 1.
• When not to use the quotient rule: a constant on top (power rule); a single term on the bottom (split it); a negative power (chain rule).
• Check a product-rule answer by expanding first when both factors are polynomials.