2.1: The Definition of the Derivative

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2.1 The Definition of the Derivative

Learning Objectives

Given a curve and a specific point, calculate the slope of the tangent line using the limit of the difference quotient, with correct application of the formula.
Identify and apply the derivative as the limit of a difference quotient to find instantaneous rates of change.

Tangent Lines

Definition $\PageIndex{1}$

Let $ f(x) $ be a function defined on an open interval containing two distinct points $ a $ and $ x $, where $ x \neq a $. The secant line to the curve $ y = f(x) $ through the points $ P(a, f(a)) $ and $ Q(x, f(x)) $ is the line passing through both points with slope

\[
m_{\text{sec}} = \frac{f(x) - f(a)}{x - a}
\]

We can also write this slope as

\[
m_{\text{sec}} = \frac{f(a + h) - f(a)}{h},
\]

where $ h \neq 0 $ and $ x = a + h $.

$alt$

Figure $\PageIndex{1}$: The slope of a secant line. From OpenStax Calculus Volume 1, licensed under CC-BY-NC-SA.

Example $\PageIndex{1}$

Find the equation of the secant line to the curve $ y = x^2 $ that passes through the points $ (3, 9) $ and $ (5, 25) $.

Definition $\PageIndex{2}$

Let $ f(x) $ be a function defined on an open interval containing $ a $. The tangent line to the curve $ y = f(x) $ at the point $ P(a, f(a)) $ is the line passing through $ P $ with slope

\[
m_{\text{tan}} = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}
\]

provided that the limit exists.

We can also express the above limit as

\[
m_{\text{tan}} = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h},
\]

where $ h \neq 0 $ and $ x = a + h $.

$This figure consists of three graphs labeled a, b, and c. Figure a shows the Cartesian coordinate plane with 0, a, x2, and x1 marked in order on the x-axis. There is a curve labeled y = f(x) with points marked (a, f(a)), (x2, f(x2)), and (x1, f(x1)). There are three straight lines: the first crosses (a, f(a)) and (x1, f(x1)); the second crosses (a, f(a)) and (x2, f(x2)); and the third only touches (a, f(a)), making it the tangent. At the bottom of the graph, the equation mtan = limx → a (f(x) - f(a))/(x - a) is given. Figure b shows a similar graph, but this time a + h2 and a + h1 are marked on the x-axis instead of x2 and x1. Consequently, the curve labeled y = f(x) passes through (a, f(a)), (a + h2, f(a + h2)), and (a + h1, f(a + h1)) and the straight lines similarly cross the graph as in Figure a. At the bottom of the graph, the equation mtan = limh → 0 (f(a + h) - f(a))/h is given. Figure c shows only the curve labeled y = f(x) and its tangent at point (a, f(a)).$

Figure $\PageIndex{2}$: The slope of tangent line. From OpenStax Calculus Volume 1, licensed under CC-BY-NC-SA.

Example $\PageIndex{2}$

Find the equation of the tangent line to the curve $ f(x) = x^2 $ at $ x = 3 $.

Example $\PageIndex{3}$

The graph of $ f(x) $ is shown below. Estimate and list all values of $ x $ where $ f(x) $ appears to have a horizontal tangent line.

$P6.png$

Figure $\PageIndex{3}$: Graph of $f(x) $

The Derivative of a Function at a Point

Definition $\PageIndex{3}$

Let $ f(x) $ be a function defined in an open interval containing $ a $. The derivative of the function $ f(x) $ at $ a $, denoted by $ f'(a) $, is defined by

\[
f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}
\]

provided that the limit exists.

We can also write the above limit as

\[
f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h},
\]

where $ h \neq 0 $ and $ x = a + h $.

Example $\PageIndex{4}$

Calculate the following derivatives using the definition of the derivative:

(a) $ f'(3) $, where $ f(x) = x^2 $

(b) $ f'(2) $, where $ f(x) = 3x^2 - 4x + 1 $

This page titled 2.1: The Definition of the Derivative was last modified on Sun, 07 Dec 2025 01:10:59 GMT and is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Kevin Palencia.

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