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\( f' \)

Derivative
Notation

Master the language of calculus: learn how to read, write, and interpret every form of derivative notation you will encounter on the AP exam.

AP Calculus AB

What Is a Derivative?

A derivative measures the instantaneous rate of change of a function at a specific point. It answers the question: "How fast is \( f(x) \) changing right now?"

Before we compute derivatives, we need to understand the notation used to express them: the symbols that tell us which function is being differentiated and with respect to which variable.

Why Notation Matters

Calculus uses multiple notation systems for derivatives. Each was invented by a different mathematician for a different purpose, and each appears on the AP exam.

Communication
Knowing every notation lets you read any textbook, any problem, any formula sheet fluently.
Flexibility
Some notations make certain operations (like the chain rule) far easier to express and remember.
AP Exam
Free-response and multiple-choice questions freely mix notation styles: you must recognize them all.

The Derivative as a Slope

x y a, f(a) tangent line slope = \( f'(a) \) \( f(x) \)

Lagrange (Prime) Notation

1
\( f'(x) \): the derivative of \( f \) as a function of \( x \).
2
\( f'(a) \): the derivative evaluated at a specific point \( x = a \).
3
\( f''(x) \): the second derivative (derivative of the derivative).

When to use it: Lagrange notation is the most compact. It is ideal when you are working with named functions like \( f \), \( g \), or \( h \) and want to keep expressions short.

Leibniz Notation

1
\( \frac{dy}{dx} \): the derivative of \( y \) with respect to \( x \). Read "dee y, dee x."
2
\( \frac{d}{dx}[\,\cdot\,] \): the operator form: "take the derivative with respect to \( x \) of whatever is inside the brackets."
3
\( \left.\frac{dy}{dx}\right|_{x=a} \): evaluate the derivative at \( x = a \).

When to use it: Leibniz notation excels at the chain rule and related rates because it explicitly shows the variable of differentiation.

Euler & Operator Notation

The capital \( D \) is called the differential operator. It means "differentiate the following expression." The subscript \( x \) clarifies the variable.

\( D_x \)
Differentiates with respect to \( x \). Equivalent to \( \frac{d}{dx} \).
\( D^2 f \)
Second derivative. Apply the operator twice: \( D(Df) = f'' \).

You will see this less often on the AP exam, but it appears in some textbooks and in higher mathematics.

Lagrange vs Leibniz

Lagrange \( f'(x) \) Leibniz \( \frac{dy}{dx} \)
Emphasis The function being differentiated The variable of differentiation
Compact? ✓ Very short Moderate length
Chain Rule \( (f \circ g)'(x) = f'(g(x))\,g'(x) \) \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \)
At a Point \( f'(3) \) \( \left.\frac{dy}{dx}\right|_{x=3} \)
Best For Quick computation Related rates, implicit differentiation

The Limit Definition in Notation

Notice how Leibniz notation hints at the definition: \( \frac{dy}{dx} \) looks like a fraction \( \frac{\Delta y}{\Delta x} \) taken to its limit. The "d" stands for an infinitesimally small change.

\( \Delta x \) \( \Delta y \) \( (x, f(x)) \) \( (x+\Delta x, f(x+\Delta x)) \) secant

As \( \Delta x \to 0 \), the secant line rotates into the tangent line, and the difference quotient becomes the derivative.

The Alternate Form of the Definition

This is equivalent to the standard form but replaces \( h \) with \( x - a \). The AP exam uses both forms interchangeably.

Standard Form
Uses \( h \to 0 \). Better for computing \( f'(x) \) as a general formula.
Alternate Form
Uses \( x \to a \). Better for finding the derivative at one specific point.

How to Read Derivative Notation

1
Identify the notation type: Look for a prime (\( ' \)), a \( \frac{d}{dx} \), or a \( D \).
2
Identify the function: What expression is being differentiated? (e.g., \( f(x) \), \( y \), \( g(t) \))
3
Identify the variable: Leibniz makes this explicit (\( dx \), \( dt \)). In Lagrange, the variable is implied from context.
4
Check for evaluation: Is there a vertical bar \( \big|_{x=a} \) or a specific number inside, like \( f'(3) \)?
5
Translate if needed: Convert between notations to pick the clearest form for the problem at hand.

Example 1: Translating Notations

Rewrite each expression in all three notations. Let \( y = f(x) = x^3 \).

Expression Lagrange Leibniz Euler
First derivative \( f'(x) = 3x^2 \) \( \frac{dy}{dx} = 3x^2 \) \( D_x f = 3x^2 \)
At \( x = 2 \) \( f'(2) = 12 \) \( \left.\frac{dy}{dx}\right|_{x=2} = 12 \) \( D_x f(2) = 12 \)
Second derivative \( f''(x) = 6x \) \( \frac{d^2y}{dx^2} = 6x \) \( D^2_x f = 6x \)

Example 2: Limit Definition

Use the limit definition to find \( f'(x) \) for \( f(x) = 5x^2 - 3x \).

1
Write the difference quotient: \( \frac{f(x+h) - f(x)}{h} \)
2
Expand \( f(x+h) \): \( 5(x+h)^2 - 3(x+h) = 5x^2 + 10xh + 5h^2 - 3x - 3h \)
3
Subtract \( f(x) \): numerator becomes \( 10xh + 5h^2 - 3h \)
4
Factor and cancel \( h \): \( \frac{h(10x + 5h - 3)}{h} = 10x + 5h - 3 \)
5
Take the limit: \( \lim_{h \to 0}(10x + 5h - 3) = 10x - 3 \)

Example 3: Alternate Form at a Point

Find \( f'(4) \) for \( f(x) = \sqrt{x} \) using the alternate definition.

1
Set up: \( f'(4) = \displaystyle\lim_{x \to 4} \frac{\sqrt{x} - \sqrt{4}}{x - 4} = \lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4} \)
2
Rationalize: Multiply top and bottom by \( \sqrt{x} + 2 \): \( \frac{(\sqrt{x}-2)(\sqrt{x}+2)}{(x-4)(\sqrt{x}+2)} = \frac{x - 4}{(x-4)(\sqrt{x}+2)} \)
3
Cancel and evaluate: \( \frac{1}{\sqrt{x}+2} \to \frac{1}{2+2} = \frac{1}{4} \)

Common Mistakes

✕ Treating \( \frac{dy}{dx} \) as a Fraction
\( \frac{dy}{dx} \) is a single symbol, not a ratio you can split arbitrarily. You cannot cancel the \( d \)'s.
✕ Forgetting the Variable
Writing \( \frac{d}{dx} \) without specifying what you are differentiating is incomplete: always include the function.
✕ Mixing Up \( f'(x) \) and \( f'(a) \)
\( f'(x) \) is a function. \( f'(a) \) is a number. Confusing them leads to errors in related-rate problems.

Notation Summary

Notation First Derivative Second Derivative At a Point
Lagrange \( f'(x) \) \( f''(x) \) \( f'(a) \)
Leibniz \( \frac{dy}{dx} \) \( \frac{d^2y}{dx^2} \) \( \left.\frac{dy}{dx}\right|_{x=a} \)
Euler \( D_x\,f \) \( D_x^2\,f \) \( D_x\,f(a) \)

Question 1 of 5

Which of the following is equivalent to \( f'(x) \)?

Question 2 of 5

What does the notation \( \left.\dfrac{dy}{dx}\right|_{x=3} \) mean?

Question 3 of 5

Using the limit definition, what is \( \displaystyle\frac{d}{dx}[3x^2] \)?

Question 4 of 5

Which limit correctly represents \( f'(a) \) using the alternate form of the definition?

Question 5 of 5

If \( g(t) = t^3 - 4t \), what is \( g'(2) \)?