The limit definition, tangent lines, and the meaning of instantaneous rate of change
The derivative is one of the two core ideas in calculus (the other is the integral). At its simplest, the derivative measures how quickly something is changing at an exact moment — like the speedometer reading in your car at a specific instant, not just the average speed over a trip.
🎯 In this section you will learn
The two limit definitions of the derivative
The geometric meaning: slope of the tangent line
The physical meaning: instantaneous rate of change
But your speed at the exact moment you pass mile marker 15 could be different — maybe you slowed down or sped up. That's your instantaneous speed. The derivative gives us a way to calculate that exact instantaneous rate of change.
📐 Geometric Meaning: Slope of the Tangent Line
On a graph, the derivative at a point is the slope of the tangent line at that point. The tangent line touches the curve at exactly one point and has the same direction as the curve at that instant.
📊 Visualizing the Derivative
Interactive graph below. The orange line is the tangent line at the selected point:
Both definitions are equivalent. The first is convenient when you only need the derivative at one specific point $x = a$. The second is often used to find the derivative function $f'(x)$.
💡 Understanding the Difference
The first definition computes the slope of the secant line between $(a, f(a))$ and $(x, f(x))$, then takes $x \to a$.
The second definition computes the slope of the secant line between $(x, f(x))$ and $(x+h, f(x+h))$, then takes $h \to 0$.
They give the same result — just two different ways of writing the same idea.
Example 1Using the Second Definition
Find $f'(x)$ for $f(x) = x^2$ using $\displaystyle f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$.
Find $f'(2)$ for $f(x) = x^2$ using $\displaystyle f'(a) = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}$.
①
Write the difference quotient with $a=2$
$\frac{f(x)-f(2)}{x-2} = \frac{x^2 - 4}{x-2}$
②
Factor the numerator
$x^2 - 4 = (x-2)(x+2)$
③
Cancel $x-2$ (for $x \neq 2$)
$\frac{(x-2)(x+2)}{x-2} = x+2$
④
Take the limit as $x \to 2$
$\lim_{x \to 2} (x+2) = 4$
⑤
Conclusion
$f'(2) = 4$ (which matches $2x$ evaluated at $x=2$)
📝 Notation for Derivatives
Notation
Meaning
$f'(x)$
"f prime of x" — derivative of $f$ with respect to $x$
$\frac{dy}{dx}$
Leibniz notation — derivative of $y$ with respect to $x$
$\frac{d}{dx}[f(x)]$
Operator notation — "the derivative of $f(x)$ with respect to $x$"
$\dot{y}$
Newton notation (used in physics for time derivatives)
⚠️ When a Derivative Does NOT Exist (Not Differentiable)
A function is differentiable at $x = a$ if $f'(a)$ exists. Common reasons a derivative might not exist:
Corner / Cusp: The function makes a sharp turn (like $f(x) = |x|$ at $x = 0$).
Discontinuity: The function has a jump or hole at $x = a$.
Vertical Tangent: The tangent line is vertical (slope infinite, like $f(x) = \sqrt[3]{x}$ at $x = 0$).
🔗 Differentiability Implies Continuity
Theorem: If $f$ is differentiable at $x = a$, then $f$ is continuous at $x = a$.
The converse is NOT true: a function can be continuous but not differentiable (like $|x|$ at $x = 0$).
🔍 Key Takeaways
The derivative $f'(x)$ is the instantaneous rate of change of $f$ at $x$.
Geometrically, $f'(a)$ is the slope of the tangent line to the graph at $x = a$.
Two equivalent definitions: $f'(a) = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}$ and $f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$.
Differentiability implies continuity — but continuity does NOT guarantee differentiability.