Derivative rules: the complete list, when each one applies, one example worked line by line, and the two rules students confuse

There are five structural derivative rules (constant, power, constant multiple, sum and difference, and the three for combinations: product, quotient, chain) and a short list of basic derivatives (xⁿ, eˣ, ln x, sin x, cos x, tan x). Every derivative you will meet in Grade 12 or first-year calculus is one of the basics wrapped in one or more of the structural rules. The skill is reading the function's structure first, then picking the rule. Here is the list, one example worked, and the two rules students mix up.

The complete list of derivative rules

Two tables. The first is the structural rules, which tell you what to do with a shape. The second is the basic derivatives, which tell you what each building block becomes. c is a constant, n is any real number, f and g are functions of x, and f' means the derivative of f.

Rule	Formula	When it applies
Constant	d/dx c = 0	A number on its own, with no x in it
Power	d/dx xⁿ = n xⁿ⁻¹	x to any real power, including negative and fractional
Constant multiple	d/dx [c f] = c f'	A number multiplying a function; the number comes along
Sum and difference	d/dx [f ± g] = f' ± g'	Terms added or subtracted; do each term alone
Product	(fg)' = f'g + fg'	Two functions multiplied side by side
Quotient	(f/g)' = (f'g − fg')/g²	One function divided by another
Chain	(f(g(x)))' = f'(g(x)) · g'(x)	One function inside another

Function	Derivative
eˣ	eˣ
aˣ	aˣ ln a
ln x	1/x
sin x	cos x
cos x	−sin x
tan x	sec²x

That is the whole list for MCV4U and for a first-year Calculus I course. If you came for a formula, it is in one of these two tables. If you came because the test is next week, keep reading: the tables are the easy part, and the marks are lost in choosing between the rows.

How to find the derivative: read the function before you pick a rule

A derivative question is a reading question first. Before you differentiate anything, ask four questions about the function in front of you, in this order.

1. Is it a sum of terms? If so, the sum rule applies: take each term on its own and add the results. Almost every polynomial starts here.
2. Is a term a constant times a power of x? Then the constant-multiple rule and the power rule do the work: the number stays, the exponent comes down in front, and the exponent drops by one.
3. Are two functions multiplied, or is one divided by another? Side by side means the product rule; one over the other means the quotient rule.
4. Is one function inside another? A bracket raised to a power, something under a root, inside a log, in an exponent, or inside sin or cos. That is the chain rule.

One step comes before all four: rewrite. The power rule only works on a term written as a power, so 1/x is x⁻¹, √x is x^(1/2), 3/x² is 3x⁻², and ∛x is x^(1/3). Most lost marks on a derivatives test are lost here, before any rule is applied, because a student tries to differentiate 1/x as it stands and has no rule that fits. Rewrite first, every time, and the four questions answer themselves.

Worked example: a sum of three terms, line by line

Find f'(x) for f(x) = 3x⁴ − 2/x + 5√x.

• Line 1, rewrite. f(x) = 3x⁴ − 2x⁻¹ + 5x^(1/2). Every term is now a constant times a power of x, so the sum rule and the power rule cover the whole thing.
• Line 2, the first term. Constant multiple and power rule: 3 · 4x³ = 12x³.
• Line 3, the second term. The exponent is −1, so bring it down and subtract one: −2 · (−1)x⁻² = 2x⁻² = 2/x². Two negatives multiplied give a positive.
• Line 4, the third term. The exponent is 1/2: 5 · (1/2)x^(−1/2) = (5/2)x^(−1/2) = 5/(2√x). The exponent dropped from 1/2 to −1/2, so the x moved to the denominator.
• Line 5, add the terms. f'(x) = 12x³ + 2/x² + 5/(2√x).

Each term was done alone and the results were added, which is all the sum rule says. Nothing in this example needed the product, quotient or chain rule, and that is the point of the four questions: the function told you which rules it wanted before you started.

The common wrong lines

Two lines a marker sees on this exact example, and why each fails.

The second term as −2/x², or as −2. The first version forgets to multiply the two negatives: the power rule gives −2 times −1, which is +2, so the term is +2/x². The second version treats 1/x as if it were x, so its derivative "is a constant". It is not. The power rule subtracts one from the exponent every time, including a negative exponent, and −1 − 1 = −2. Writing 1/x as x⁻¹ before you differentiate makes both slips visible.

The third term as (5/2)√x. The exponent went up instead of down. The power rule says 1/2 − 1 = −1/2, and a negative exponent means the x belongs in the denominator: 5/(2√x). A quick check catches it: the derivative of a root should get smaller as x grows, because √x flattens out, and (5/2)√x grows.

A sibling to try cold

Cover the worked example and find g'(x) for g(x) = 4x³ + 6/x² − 2√x. Rewrite first, then one term at a time. Write every exponent down; do not do the subtraction in your head.

Check

g'(x) = 12x² − 12/x³ − 1/√x.

If your second term came out positive, the two negatives were not multiplied: 6 · (−2) = −12. If your third term has √x on top, the exponent went the wrong way. Either miss is a card for tonight, with the rewrite written on the front.

The two differentiation rules students confuse: product and chain

Here are two functions that look alike and want different rules.

(x² + 1)(x³) is a product: two functions standing beside each other, multiplied. (x² + 1)³ is a composition: one function, x² + 1, sitting inside another, the cube. The tell is one question: inside, or beside? If you can point at an inside, it is the chain rule. If the two pieces stand side by side, it is the product rule.

The product, worked: f = x² + 1, g = x³, so f' = 2x and g' = 3x². Then f'g + fg' = 2x · x³ + (x² + 1) · 3x² = 2x⁴ + 3x⁴ + 3x² = 5x⁴ + 3x². Check it by expanding first: (x² + 1)(x³) = x⁵ + x³, whose derivative by the power rule is 5x⁴ + 3x². Same answer, which is what the check is for.

The composition, worked: the outside is a cube, the inside is x² + 1, and the inside's derivative is 2x. The chain rule gives 3(x² + 1)² · 2x = 6x(x² + 1)². The last factor, 2x, is the one students drop; the chain rule, worked is a whole page on that one factor, and the product and quotient rules, worked carries the sign error on the quotient side.

The second confusion is smaller and costs the same mark. The derivative of sin x is cos x, with no sign change. The derivative of cos x is −sin x. The minus sign lives on cosine's derivative only. If you remember one fact from the trig column of the table, remember that one.

What the derivative rules are not for

Three places where a rule looks like it applies and does not.

• A constant that looks like a power. π² is a number. Its derivative is 0, not 2π. The power rule is for x to a power, not for a number to a power.
• xˣ. This is neither a power of x (the exponent is not constant) nor an exponential (the base is not constant). It needs logarithmic differentiation, which is a later idea and is out of scope here; the table has no row for it on purpose.
• A quotient with a constant on top. 5/x² does not need the quotient rule: write it as 5x⁻² and use the power rule. More generally, 1/g is g⁻¹, and the chain rule handles it with no subtraction order to get wrong. The quotient rule is for a changing top over a changing bottom that will not split.

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

In MCV4U these rules are five ideas in a row: idea 6, Power rule; derivatives of polynomials, then idea 7, Product and quotient rules, then idea 8, Chain rule, and after idea 9 (implicit differentiation) come idea 10, Derivatives of exponential and logarithmic functions, and idea 11, Derivatives of trigonometric functions. Before them is idea 5, The derivative from first principles, which is where the power rule comes from. The MCV4U course map shows all 24 ideas in teaching order.

In a first-year Calculus I course the same material is idea 5, Power, product and quotient rules, and idea 6, The chain rule, after idea 4, The derivative as a limit, with the exponential, log and trig derivatives at ideas 8 and 9. First-year calculus, what changes is written for the Grade 12 student about to meet it.

What comes next uses every row of the table. Implicit differentiation is the chain rule with y as the inside. Extrema, concavity and curve sketching (MCV4U ideas 13 to 15) are these derivatives set equal to zero and read. And the skill this page is about, choosing the rule, is taught by one kind of practice: mixed. Rohrer and Taylor (2007) found that students who practised problem types shuffled together did worse during practice and better on the test than students who practised in blocks, because a block tells you the rule before you read the problem, and a shuffled set makes you read. Rohrer, Dedrick and Stershic (2015) found the same effect in a classroom over months. Why mixed practice teaches the choice of rule has the method; worked examples, then practice problems has the order to do it in. Sweller and Cooper (1985) is the reason the example above comes before the sibling.

Derivative rules on one screen

• Structural rules: constant → 0; xⁿ → n xⁿ⁻¹; c f → c f'; f ± g → f' ± g'; (fg)' = f'g + fg'; (f/g)' = (f'g − fg')/g²; (f(g(x)))' = f'(g(x)) · g'(x).
• Basics: eˣ → eˣ; aˣ → aˣ ln a; ln x → 1/x; sin x → cos x; cos x → −sin x; tan x → sec²x.
• Rewrite first: 1/x = x⁻¹, √x = x^(1/2), 3/x² = 3x⁻².
• The four questions: a sum of terms? a constant times a power? multiplied or divided? one inside another?
• The two confusions: inside means chain, beside means product; the minus sign is on cosine's derivative only.
• Not for: a constant like π² (derivative 0); xˣ (logarithmic differentiation, later); a constant over a power (rewrite, power rule).