Calculus – Differential Calculus

Overview

Differential calculus is the part of calculus that focuses on change. It answers questions like:

• How fast is something moving right now?
• How quickly is a quantity increasing or decreasing at a specific moment?
• What is the slope of a curve at a single point?

The central tool of differential calculus is the derivative. In this chapter, you will see what derivatives are about, how they arise from limits, and what kinds of problems they help solve. Precise definitions and techniques will be developed in the later chapters on limits and derivatives; here the goal is to understand the big picture of differential calculus as a whole.

The Central Idea: Instantaneous Rate of Change

In everyday life, you often deal with average rates of change:

• A car travels 120 km in 2 hours. Its average speed is $60$ km/h.
• A population grows from 1,000 to 1,200 in 5 years. The average growth is $40$ individuals per year.

An average rate of change of a quantity $y$ over a time interval from $t=a$ to $t=b$ can be written using a function $y=f(t)$ as
$$
\text{average rate of change} = \frac{f(b)-f(a)}{b-a}.
$$

Differential calculus asks for something more precise: not over a whole interval, but at a single moment. This is the instantaneous rate of change.

You can think of it this way:

• Average speed between 1 s and 2 s can be computed easily.
• Instantaneous speed at exactly $t=1$ s is what the car’s speedometer shows. It cannot be found by a simple division; it requires a limiting process, which is the subject of the limits chapter.

In differential calculus, we interpret the instantaneous rate of change of $y=f(x)$ at $x=a$ as the slope of the curve $y=f(x)$ at that point.

Tangent Lines and Slopes

In basic algebra and geometry, the slope of a straight line is a single number that tells you how steep the line is. For a line through $(x_1,y_1)$ and $(x_2,y_2)$, the slope is
$$
m = \frac{y_2-y_1}{x_2-x_1}.
$$

Curves are more complicated: their steepness changes from point to point. Near a fixed point $x=a$, we can approximate the curve by a line that

• touches the curve at $(a,f(a))$, and
• best matches the behavior of the curve near that point.

This line is called the tangent line at $x=a$. Its slope is, by definition, the derivative of $f$ at $a$.

The big idea:

• On a graph, the derivative at a point is the slope of the tangent line at that point.
• Numerically, the derivative is the instantaneous rate of change of the function at that point.

Later, when you formally define derivatives, you will see that they come from slopes of secant lines (lines through two points of the curve) by shrinking the distance between the points using limits.

Derivatives as Functions

For a function $y=f(x)$, you can often find the derivative at every point where it makes sense. This gives you a new function, written in various ways:

• $f'(x)$ (read “$f$ prime of $x$”),
• $\dfrac{dy}{dx}$ (read “dy dx” or “the derivative of $y$ with respect to $x$”).

This new function $f'(x)$ tells you, for each $x$:

• how $f(x)$ is changing at that point,
• the slope of the tangent line to the graph of $y=f(x)$ at $x$.

So, differential calculus is not just about computing a single number; it is about constructing a whole new function that encodes rates of change.

Local Behavior of Functions

Differential calculus connects derivatives to how a function behaves locally (in a small neighborhood of a point).

Some key qualitative ideas:

• If $f'(x) > 0$ near a point, $f$ is increasing there.
• If $f'(x) < 0$ near a point, $f$ is decreasing there.
• Points where $f'(x)=0$ are candidates for local maxima or minima (peaks or valleys).

Although the detailed rules for using derivatives to study graphs appear in the chapter on applications of derivatives, you should already view the derivative as a tool that:

• tells you where a function goes up or down,
• indicates where the function might change from increasing to decreasing, or vice versa.

This is the foundation for using calculus in optimization and curve sketching.

Linear Approximation

A subtle but very important idea in differential calculus is that smooth functions behave almost like straight lines when you zoom in close enough.

Near $x=a$, if $f$ is differentiable, then you can approximate $f(x)$ by a straight line:
$$
f(x) \approx f(a) + f'(a)(x-a)
$$
for $x$ close to $a$.

This is called a linear approximation (or first-order approximation). It means:

• $f(a)$ gives the value at the center point,
• $f'(a)(x-a)$ adjusts that value based on how far you move from $a$ and how fast $f$ is changing there.

Conceptually, linear approximations allow you to:

• replace complicated behavior by a simple line, temporarily, in a small region,
• estimate values of functions without exact calculations,
• set up more advanced ideas like error estimates and series expansions (these go beyond this course level but are built on the same idea).

Differential Notation

In differential calculus, you encounter symbols like $dy$, $dx$, and $\dfrac{dy}{dx}$. While the formal treatment of these symbols belongs to the derivatives chapter, it is helpful to understand their role at a high level.

• $dx$ represents a small change in the input $x$.
• $dy$ represents the corresponding change in the output $y$.
• $\dfrac{dy}{dx}$ represents the ratio of these small changes, which is the derivative.

For a differentiable function $y=f(x)$, when $dx$ is small,
$$
dy \approx f'(x)\,dx.
$$

This is a restatement of the linear approximation idea in differential language, and it will be used in practical applications such as related rates and differentials.

Conceptual Types of Problems in Differential Calculus

Differential calculus is organized around a few broad types of tasks. Later chapters focus on each type in more detail; here is how they fit together conceptually.

1. Finding Derivatives

Given a function $f(x)$, you will learn to:

• define its derivative using a limit,
• then use systematic rules to compute derivatives without repeating the limit process each time.

This is the computational backbone of differential calculus: turning a function into its derivative function.

2. Using Derivatives to Optimize

In many real-world situations, you want to make something as large or as small as possible:

• maximize profit,
• minimize cost,
• minimize distance,
• maximize area, and so on.

Differential calculus provides a general approach:

• find where the derivative is zero or does not exist,
• interpret these special points to determine maxima and minima, using additional reasoning or tests.

The chapter on applications of derivatives will explore this in depth, but conceptually, differential calculus turns “best choice” problems into questions about where the derivative vanishes or changes sign.

3. Analyzing Motion and Change

When a quantity changes over time—position, temperature, concentration—the derivative describes how fast it is changing:

• velocity is the derivative of position with respect to time,
• acceleration is the derivative of velocity (and the second derivative of position).

Differential calculus allows you to:

• translate a description of motion into a function and its derivatives,
• interpret what the derivative tells you about speeding up, slowing down, and direction changes.

This underlies many physical and engineering applications.

4. Understanding Curves and Their Shapes

Derivatives describe not only whether a function is increasing or decreasing, but also how its rate of change itself is changing. Using first and second derivatives, you can:

• understand the shape of a graph,
• locate peaks and valleys,
• find points where the curvature changes.

These techniques support more accurate graph sketching and qualitative understanding of functions without plotting many points.

How Differential Calculus Relates to Integral Calculus

This course separates calculus into differential and integral parts, but they are deeply connected.

• Differential calculus moves from functions to derivatives (rates of change, slopes).
• Integral calculus moves from derivatives back to functions, often through accumulation.

The link between the two is formalized by the Fundamental Theorem of Calculus (studied later). For now, you only need the broad picture:

• If a function measures something like position, its derivative gives velocity.
• If you know the velocity function, integration can recover the original position (up to a constant).

Differential calculus is usually studied first because:

• you must understand derivatives and rates of change
• before you can fully appreciate accumulation as the “inverse process” in integral calculus.

What You Will Learn in the Differential Calculus Part of the Course

The chapters grouped under “Calculus – Differential Calculus” develop this subject in stages:

• Limits and Continuity: the precise language for talking about approaching a value and for deciding when a function behaves nicely enough to have a derivative.
• Derivatives: formal definitions and practical rules for computing derivatives of many kinds of functions.
• Applications of Derivatives: using derivatives to solve problems in optimization, related rates, and curve sketching.

In this broader chapter, your main takeaway should be:

• Differential calculus is about using derivatives to understand and control change.
• The derivative links algebraic formulas, geometric pictures (slopes and tangents), and real-world quantities (speed, growth, and more).
• Much of modern science, engineering, and many areas of mathematics depend on the concepts that grow out of this central idea.