Limit Definition of the Derivative
In short: Limit definition of the derivative builds the exact slope of a curve by shrinking the gap h between two points toward zero and watching the secant slopes close in on one value.
AP Calculus ABCalculusAP Calculus AB Unit 2
What is Limit Definition of the Derivative?
The limit definition of the derivative turns a skill you already have, finding the slope between two points, into a tool for exact steepness. Mark one spot on the graph, mark a neighbor \(h\) units to the right, and compute the slope of the line through both.
Now squeeze the neighbor toward the first spot by letting \(h\) go to 0. When those slopes home in on a single number, the result is the derivative:
\[f^{\prime}(x) = \lim_{h \to 0} \frac{f(x + h) – f(x)}{h}\]
An equivalent form at a fixed point uses \(x \to a\): \(f^{\prime}(a) = \lim_{x \to a} \frac{f(x) – f(a)}{x – a}\).
How to spot it
- The problem says “use the definition of the derivative” or “find f'(x) from first principles”.
- An expression like (f(x+h) – f(x))/h or (f(x) – f(3))/(x – 3) appears and a limit must be evaluated.
- A rule such as the power rule is not allowed or not yet known, so the limit must be done by hand.
- You must decide whether a function has a derivative at a sharp corner or a break.
The method
- Write \(f(x + h)\) by replacing every \(x\) with \(x + h\), and form \(\frac{f(x+h) – f(x)}{h}\).
- Expand and simplify the numerator until every term has a factor of \(h\).
- Cancel \(h\) from the top and bottom; direct substitution of \(h = 0\) now works.
- Let \(h \to 0\) and read off \(f^{\prime}(x)\).
Limit Definition of the Derivative: solved example
Problem
Use the limit definition of the derivative to find \(f^{\prime}(x)\) for \(f(x) = 4x^2 – 7x\).
Solution
Compute \(f(x+h) = 4(x+h)^2 – 7(x+h) = 4x^2 + 8xh + 4h^2 – 7x – 7h\). Subtract \(f(x)\): the numerator becomes \(8xh + 4h^2 – 7h\).
Divide by \(h\): \(\frac{8xh + 4h^2 – 7h}{h} = 8x + 4h – 7\). Now let \(h \to 0\): \(f^{\prime}(x) = 8x – 7\).
Try it
Question 1
Use \(\lim_{h \to 0} \frac{f(x+h) – f(x)}{h}\) to find \(f^{\prime}(x)\) when \(f(x) = 5x + 9\).
Show the answer
The numerator is \(5(x+h) + 9 – 5x – 9 = 5h\). Dividing by \(h\) gives 5, so \(f^{\prime}(x) = 5\).
Question 2
For \(f(x) = x^2\), evaluate \(\lim_{x \to 3} \frac{f(x) – f(3)}{x – 3}\).
Show the answer
\(\frac{x^2 – 9}{x – 3} = x + 3\) for \(x \neq 3\), and the limit is \(3 + 3 = 6\), which is \(f^{\prime}(3)\).
Limit Definition of the Derivative vs. Derivative Rules
The definition is the source: it works for any function but takes algebra every time. The shortcut rules, such as the power rule and the product rule, are theorems proved from this limit. Use the definition when a problem asks for it by name; use the rules when you only need the answer.
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