3.1 Defining the Derivative

The Derivative and Tangent Lines

Derivatives answer a fundamental question: how fast is something changing at a specific instant? They give you the slope of a curve at any point, which turns out to be useful for analyzing motion, rates of change, and optimization problems. This section covers how derivatives are defined through limits and how they connect to tangent lines and velocity.

Tangent Line Concept

A tangent line is a straight line that just touches a curve at a single point (called the point of tangency) and matches the curve's direction at that point. Think of it as the best straight-line approximation of the curve right at that spot.

The slope of the tangent line tells you the rate of change of the curve at that point:

• Positive slope: the curve is increasing
• Negative slope: the curve is decreasing
• Zero slope: the curve has a horizontal tangent (a local max, min, or inflection point could be here)

The tangent line is closely related to the secant line, which passes through two points on the curve. As you slide those two points closer together, the secant line approaches the tangent line. That limiting process is exactly how we define the derivative.

Tangent Slope Calculation

To find the slope of the tangent line at a point $x_0$, you take the limit of the difference quotient:

$m_{\text{tangent}} = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}$

The difference quotient $\frac{f(x_0 + h) - f(x_0)}{h}$ represents the average rate of change of $f$ over the interval $[x_0,\, x_0 + h]$. That's just the slope of the secant line through $(x_0, f(x_0))$ and $(x_0 + h, f(x_0 + h))$.

As $h \to 0$, the interval shrinks, the secant line rotates toward the tangent line, and the average rate of change becomes the instantaneous rate of change.

Derivative as a Limit

The derivative of a function $f(x)$ at a point $x_0$ is formally defined as:

$f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}$

provided this limit exists. When you leave $x_0$ as a variable $x$, you get the derivative function $f'(x)$, which gives the instantaneous rate of change at any point in the domain.

Common notations for the derivative:

• Prime notation: $f'(x)$
• Leibniz notation: $\frac{df}{dx}$ or $\frac{d}{dx}f(x)$

Both mean the same thing. Leibniz notation is especially handy when you want to emphasize what variable you're differentiating with respect to.

Derivative at a Specific Point

To compute the derivative at a specific point, say $x_0 = 3$:

1. Write out the difference quotient: $\frac{f(3 + h) - f(3)}{h}$

2. Substitute $f(3 + h)$ and $f(3)$ using the function's formula, then simplify algebraically (factor, cancel the $h$ in the denominator, etc.)

3. Evaluate the limit as $h \to 0$

The result is both the slope of the tangent line at $x_0 = 3$ and the instantaneous rate of change of $f$ at that point.

Continuity and Differentiability

Continuity is necessary but not sufficient for differentiability. If a function isn't continuous at a point, it can't be differentiable there. But a function can be continuous at a point and still fail to be differentiable.

Three common situations where continuity holds but differentiability fails:

• Sharp corners or cusps: the curve changes direction abruptly (e.g., $f(x) = |x|$ at $x = 0$)
• Vertical tangent lines: the slope approaches $\pm \infty$
• Discontinuities: jumps, holes, or asymptotes automatically rule out differentiability

The key takeaway: differentiability implies continuity, but continuity does not imply differentiability.

Derivatives and Velocity

Velocity as Rate of Change

Velocity is the most natural real-world example of a derivative. If $s(t)$ describes an object's position at time $t$, then the derivative $s'(t)$ gives the instantaneous velocity at time $t$.

• $s'(t) > 0$: the object is moving in the positive direction
• $s'(t) < 0$: the object is moving in the negative direction
• $s'(t) = 0$: the object is momentarily at rest

Average vs. Instantaneous Velocity

Average velocity measures displacement over a time interval:

$v_{\text{avg}} = \frac{\Delta s}{\Delta t} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}$

This is the slope of the secant line on the position-time graph between $t_1$ and $t_2$.

Instantaneous velocity is the limit of average velocity as the time interval shrinks to zero:

$v_{\text{inst}} = \lim_{\Delta t \to 0} \frac{\Delta s}{\Delta t} = \frac{ds}{dt}$

Notice the parallel: average velocity uses a secant line, instantaneous velocity uses a tangent line. The derivative bridges the two.

Derivative Estimation from Data

You won't always have a nice formula. Sometimes you'll need to estimate a derivative from a table or graph.

From a table:

1. Pick two points close to your point of interest: $(x_1, y_1)$ and $(x_2, y_2)$

2. Compute the average rate of change: $\frac{y_2 - y_1}{x_2 - x_1}$

3. For a better estimate, use points on both sides of your target and average the two slopes (this is called a central difference estimate)

For example, if you want $f'(3)$ and your table has values at $x = 2, 3, 4$, compute $\frac{f(4) - f(2)}{4 - 2}$ for a centered estimate.

From a graph:

1. Sketch a tangent line to the curve at the point of interest
2. Pick two clear points on that tangent line (use grid intersections when possible)
3. Calculate rise over run between those two points to estimate the slope