Prime Notation — Definition, Formula & Examples
Prime notation is a way of writing derivatives by placing a prime mark (an apostrophe-like tick) after the function name. For example, if is a function, then represents its first derivative with respect to .
In Lagrange's notation, the th derivative of a function is denoted , where is the first derivative, is the second derivative, and so on. Each prime mark indicates one additional application of the differentiation operator with respect to the function's independent variable.
Key Formula
Where:
- = The first derivative of f, written in prime notation
- = The original function
- = A small increment approaching zero
How It Works
You start with a function name like , , or and add tick marks to show how many times you have differentiated. One tick gives the first derivative: . Two ticks give the second derivative: . For derivatives beyond the third, writing individual tick marks becomes awkward, so you switch to a numeral in parentheses: rather than . Prime notation is compact and especially convenient when you are evaluating a derivative at a specific point, such as .
Worked Example
Problem: Let f(x) = 3x^4. Find f'(x) and f''(x) using prime notation, then evaluate f'(2).
Find f'(x): Apply the power rule to differentiate once.
Find f''(x): Differentiate f'(x) to get the second derivative.
Evaluate f'(2): Substitute x = 2 into the first derivative.
Answer: , , and .
Why It Matters
Prime notation is the default notation on AP Calculus exams (both AB and BC). Multiple-choice and free-response questions routinely use , , and , so fluency with these symbols is essential for reading problems quickly and correctly.
Common Mistakes
Mistake: Confusing the prime mark with an exponent, reading f'(x) as f raised to the first power.
Correction: The prime mark sits after the function name, not as a superscript on a variable. means the derivative of , while would just equal .