The Derivative & Basic Rules

You’ll be able to

Define the derivative as a limit of a difference quotient
Apply the power, constant-multiple, and sum rules
Differentiate products and quotients with the product and quotient rules

The derivative is an instantaneous rate

The derivative f′(x) measures how fast f changes at a single instant — the slope of the tangent line. It is defined as the limit of a difference quotient, the slope of a secant line as its two points squeeze together. Geometrically the secant becomes the tangent; physically an average rate becomes an instantaneous rate. A function is differentiable at a point only if this limit exists, which requires the graph to be continuous and smooth (no corners or vertical tangents) there.

Limit definition of the derivative

f′(x) = lim(h→0) [f(x + h) − f(x)] / h

The difference quotient [f(x+h) − f(x)]/h is the secant slope; its limit as h→0 is the tangent slope.

The workhorse: the power rule

For f(x) = xⁿ, the derivative is f′(x) = n·xⁿ⁻¹ — bring the exponent down as a coefficient and drop it by one. This holds for any real n, including negatives and fractions, so 1/x = x⁻¹ and √x = x^(1/2) are both handled by rewriting first. Combine it with the constant-multiple rule (constants ride along: (c·f)′ = c·f′) and the sum rule (differentiate term by term) to attack any polynomial instantly.

Product and quotient rules

(fg)′ = f′g + fg′ · (f/g)′ = (f′g − fg′) / g²

The quotient rule numerator is "low d-high minus high d-low"; the order of subtraction matters.

Worked example

Differentiate y = x² · sin(x).

1.This is a product of f = x² and g = sin(x), so use the product rule (fg)′ = f′g + fg′.
2.Differentiate each factor: f′ = 2x and g′ = cos(x).
3.Assemble: y′ = (2x)(sin x) + (x²)(cos x).
4.Write it cleanly: y′ = 2x·sin(x) + x²·cos(x).

Answer: y′ = 2x·sin(x) + x²·cos(x). Both terms are kept — a product does not differentiate factor-by-factor.

Watch out

The derivative of a product is not the product of the derivatives. (fg)′ ≠ f′g′. You must use the product rule; forgetting the second term (fg′) is one of the most frequent Unit 2 errors.

Checkpoint

What is the derivative of f(x) = 4x³ − 2x + 7?

Checkpoint

Using the quotient rule, what is the derivative of y = x / (x + 1)?

On the exam

When a limit is written as lim(h→0) [f(a+h) − f(a)]/h, recognize it as the definition of a derivative at a point, f′(a). AP loves disguising a derivative as a raw limit — identifying it lets you skip the algebra and just differentiate.

Answer the 2 checkpoints as you read.