The limit definition of a derivative for a function $f(x)$is as follows:

\[ \begin{align*} \lim_{h\to0} \dfrac{f(x+h) – f(x)}{h} \end{align*} \]

To understand this function, imagine two points. Both are on the $f(x)$ curve, and one is at $(x,f(x))$, while the other is at $(x+h,f(x+h)$. What this function says as the second point approaches the first point, the slope between them will be the derivative at $x$.

As an example, lets look at $f(x) = x^2 + 2x$. Using the definition, we get:

\[ \begin{align*} f'(x) &= \lim_{h\to0} \dfrac{f(x+h) – f(x)}{h}\\ \dfrac{d}{dx} (x^2 + 2x) &= \lim_{h\to0} \dfrac{(x + h)^2 + 2(x + h) – (x^2 + 2x)}{h}\\ &= \lim_{h\to0} \dfrac{x^2 + 2xh + h^2 + 2x + 2h – x^2 – 2x}{h}\\ &= \lim_{h\to0} \dfrac{h^2 + 2xh + 2h}{h}\\ &= \lim_{h\to0} h + 2x + 2 = 2x + 2 \end{align*} \]

We get that the derivative of $f(x) = x^2 + 2x$ is $\boxed{2x + 2}$.

You are here:

- Home
- Derivatives
- Limit Definition of Derivative

Master Calculus! Get instant help on “I Aced Calculus AP” App. Hundreds of flashcards and practice questions at your fingertips. Download now on the App Store and Google Play.

# Limit Definition of Derivative

- I Aced Calculus Team
- October 7th, 2024
- Categories: Derivatives