Difficulty: Introductory-Intermediate | Prerequisites: Limits (Unit 1), basic algebra, trigonometric identities.
Derivatives are the central tool of differential calculus. They measure how a function changes at a single point, giving you the slope of the tangent line to a curve rather than the slope of a secant line between two points. If Unit 1 was about what a function approaches, Unit 2 is about what a function is doing right now, at one instant.
You will use derivatives constantly from here on: in optimisation, in related rates, in curve sketching, and on the AP exam. Every rule in this set of notes is a building block for those later topics, so solid fluency here saves you time everywhere else.
The derivative of a function at a point is the limit of the difference quotient. Several shortcut rules (power, product, quotient, chain) let you skip the limit process for common function types. Derivatives of trigonometric functions follow their own patterns and combine freely with those rules.
Derivative (limit definition)
The derivative of f at x is defined as lim(h→0) [f(x+h) − f(x)] / h. It gives the slope of the tangent line to the curve at any point x.
In simple terms, it is the instantaneous rate of change of a function, the slope at a single point rather than between two points.
Alternate form of the derivative
lim(x→c) [f(x) − f(c)] / (x − c). A rearrangement of the limit definition that is handy when you are given a specific x-value c and want the slope there directly.
Think of it as: plug in two versions of x (the variable one and the fixed one), then take the limit as the variable collapses onto the fixed point.
Differentiable
A function is differentiable at a point if its derivative exists there. Differentiability at x = c guarantees continuity at x = c, but not the other way round.
In simple terms, if you can draw a single, well-defined tangent line at a point, the function is differentiable there.
Tangent line
A line that touches a curve at exactly one local point and has the same slope as the curve at that point. Its slope equals the derivative evaluated there.
Think of it as: the best straight-line approximation of the curve at that instant.
The formal (limit) definition: f'(x) = lim(h→0) [f(x+h) − f(x)] / h
This computes the slope of the tangent line to y = f(x) at any point.
Example: y = x² + 1 at x = 2.
lim(h→0) [(x+h)² + 1 − (x² + 1)] / h = lim(h→0) [2xh + h²] / h = 2x.
At x = 2: f'(2) = 4.
f'(c) = lim(x→c) [f(x) − f(c)] / (x − c)
Useful as a shortcut when you already know the particular x-value you care about.
Example: f(x) = x² − 5, find f'(3).
lim(x→3) (x² − 5 − 4) / (x − 3) = lim(x→3) (x² − 9) / (x − 3) = lim(x→3) (x+3)(x−3) / (x−3) = 6.
A derivative does not exist at a point where the function has:
A corner (sharp change in direction, like |x| at x = 0)
A cusp (both sides approach the point but with slopes going to ±∞)
A vertical tangent (the slope is infinite)
A discontinuity (the function jumps or is undefined)
If f is differentiable at x = c, then f is continuous at x = c. The converse is false: a function can be continuous at a point but not differentiable there (corners, cusps, vertical tangents).
For y = xⁿ, the derivative is y' = n·xⁿ⁻¹.
Example: y = 5x², so y' = 10x.
Works for any real exponent, including negatives and fractions.
For y = f(x)·g(x): y' = f(x)·g'(x) + g(x)·f'(x).
In words: first times derivative of second, plus second times derivative of first.
For y = f(x)/g(x): y' = [g(x)·f'(x) − f(x)·g'(x)] / [g(x)]².
In words: bottom times derivative of top, minus top times derivative of bottom, all over bottom squared.
For y = f(g(x)): y' = f'(g(x))·g'(x).
Differentiate the outer function, leave the inner function alone, then multiply by the derivative of the inner function.
Example: f(x) = (x+2)³, so f'(x) = 3(x+2)²·1 = 3(x+2)².
Example with trig: f(x) = sin³(4x).
f'(x) = 3·sin²(4x)·cos(4x)·4 = 12 sin²(4x) cos(4x).
d/dx [sin x] = cos x
d/dx [cos x] = −sin x
d/dx [tan x] = sec² x
These combine with the chain rule whenever the argument is something other than plain x.
Rule | Formula |
|---|---|
Power Rule | d/dx [xⁿ] = n·xⁿ⁻¹ |
Product Rule | d/dx [f·g] = f·g' + g·f' |
Quotient Rule | d/dx [f/g] = (g·f' − f·g') / g² |
Chain Rule | d/dx [f(g(x))] = f'(g(x))·g'(x) |
sin x | cos x |
cos x | −sin x |
tan x | sec² x |
Derivatives describe any quantity that changes: velocity is the derivative of position, acceleration is the derivative of velocity. In engineering and physics, every "rate" you encounter is a derivative in disguise. In economics, marginal cost and marginal revenue are derivatives of total cost and total revenue functions.
Students frequently forget to multiply by the inner derivative when using the chain rule. If the argument is anything other than plain x, the chain rule applies.
The quotient rule order matters: it is (bottom × derivative of top) minus (top × derivative of bottom), not the other way round. Swapping the subtraction flips the sign of the entire answer.
Differentiability implies continuity, but continuity does not imply differentiability. The absolute value function at x = 0 is the classic counterexample.
L'Hôpital's Rule (covered in Part 2) is not the quotient rule. They look superficially similar but do completely different things.
⚠️ The limit definition of the derivative appears regularly on both multiple-choice and free-response sections. You must recognise it in both its standard and alternate forms.
⚠️ Knowing exactly when a derivative does not exist (corners, cusps, vertical tangents, discontinuities) is a common conceptual question.
⚠️ Chain rule fluency is non-negotiable. Most derivative problems on the AP exam require at least one application of the chain rule.
⚠️ Trig derivatives appear constantly in combination with product, quotient, and chain rules. Memorise the three core trig derivatives cold.
True or False: If a function is continuous at a point, it must be differentiable there.
Fill in the blank: The derivative of sin(x) is ______.
True or False: The chain rule says to differentiate the outer function and multiply by the derivative of the inner function.
Fill in the blank: For y = xⁿ, the derivative y' = ______.
True or False: A function with a vertical tangent at x = 3 has a derivative at x = 3.
Answers: 1. False. 2. cos(x). 3. True. 4. n·xⁿ⁻¹. 5. False.
Q: Using the limit definition, find the derivative of f(x) = x² + 1.
A: f'(x) = lim(h→0) [(x+h)² + 1 − (x² + 1)] / h = lim(h→0) [2xh + h²] / h = 2x.
Q: Using the alternate form of the derivative, find f'(3) for f(x) = x² − 5.
A: lim(x→3) (x² − 9)/(x − 3) = lim(x→3) (x+3) = 6.
Q: Differentiate f(x) = (x + 2)³ using the chain rule.
A: f'(x) = 3(x + 2)² · 1 = 3(x + 2)².
Q: Differentiate f(x) = sin³(4x).
A: f'(x) = 3 sin²(4x) · cos(4x) · 4 = 12 sin²(4x) cos(4x).
Q: Name the four cases where a derivative does not exist.
A: Corners, cusps, vertical tangents, and discontinuities.
This material connects directly to Unit 1 (Limits), since the derivative is itself a limit. It also sets the stage for Unit 3 topics like curve sketching and optimisation, which rely entirely on first and second derivatives. Related rates and implicit differentiation (covered in Part 2 of these notes) are direct applications of the chain and product rules introduced here.
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