Introduction to Derivatives

Welcome to derivatives, students 👋

Calculus helps us describe change. In the real world, change is everywhere: a car speeds up, a phone battery drains, a plant grows, and a ball falls after being thrown. In IB Mathematics Analysis and Approaches SL, derivatives are one of the main tools used to study change. This lesson introduces the core idea behind derivatives and explains why they are so useful in calculus.

Learning objectives

By the end of this lesson, students, you should be able to:

explain the main ideas and terminology behind derivatives;
use basic IB reasoning and procedures connected to derivatives;
connect derivatives to the wider study of calculus;
describe how derivatives fit into applications such as gradients, motion, and optimization;
use examples to show what a derivative means in context.

A derivative is often described as the rate of change of a function. It also tells us the slope of a curve at a specific point. This is powerful because many real situations are not straight lines, so we need a way to measure change on a curve instead of only on a line 📈.

From average change to instant change

Before learning the formal derivative, it helps to start with average rate of change. Suppose a runner’s distance from the start is given by a function $s(t)$, where $t$ is time. Over a time interval from $t=a$ to $t=b$, the average rate of change is

$$\frac{s(b)-s(a)}{b-a}$$

This formula gives the slope of the straight line joining the two points on the graph. That line is called a secant line.

For example, if $s(t)=t^2$ and we compare $t=2$ and $t=5$, the average rate of change is

$$\frac{5^2-2^2}{5-2}=\frac{25-4}{3}=7$$

So, on average, the runner’s distance is increasing by $7$ units per second over that interval.

But what if we want the rate of change at exactly one moment, such as $t=3$? That is where the derivative comes in. Instead of using two points far apart, we make the interval smaller and smaller until it becomes a single point. This idea leads to the instantaneous rate of change.

The derivative as a limit

The formal definition of the derivative uses a limit. If a function is $f(x)$, then its derivative at $x$ is written as $f'(x)$ or $\frac{dy}{dx}$ when $y=f(x)$. The definition is

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

This expression measures how much the function changes when the input changes by a very small amount $h$.

Why is the limit important? Because the fraction

$$\frac{f(x+h)-f(x)}{h}$$

is an average rate of change over a tiny interval of width $h$. As $h$ gets closer to $0$, the interval shrinks, and we get the rate of change at a single point.

Let’s use $f(x)=x^2$ as an example. Then

$$f'(x)=\lim_{h\to 0}\frac{(x+h)^2-x^2}{h}$$

Expanding the bracket gives

$$f'(x)=\lim_{h\to 0}\frac{x^2+2xh+h^2-x^2}{h}$$

Simplifying,

$$f'(x)=\lim_{h\to 0}\frac{2xh+h^2}{h}=\lim_{h\to 0}(2x+h)$$

Now let $h\to 0$:

$$f'(x)=2x$$

This means the gradient of the curve $y=x^2$ changes depending on $x$. At $x=3$, the gradient is $6$. At $x=-2$, the gradient is $-4$.

What the derivative means on a graph

The derivative has two closely related meanings:

Gradient of the tangent line to the graph at a point.
Instantaneous rate of change of the function at that point.

A tangent line touches a curve at one point and has the same gradient as the curve at that point. If a curve is rising, the derivative is positive. If it is falling, the derivative is negative. If the curve is flat, the derivative is zero.

These facts help us interpret graphs quickly:

$f'(x)>0$ means $f(x)$ is increasing;
$f'(x)<0$ means $f(x)$ is decreasing;
$f'(x)=0$ may mean a turning point or a flat section.

For instance, if the height of a ball is modeled by a function $h(t)$, then $h'(t)$ tells us the vertical velocity at time $t$. When the ball reaches its highest point, its velocity is momentarily $0$. That is why the derivative is so useful in kinematics 🚀.

Derivative notation and terminology

IB Mathematics uses several notations for derivatives. It is important to recognize all of them:

$f'(x)$: the derivative of $f$ with respect to $x$;
$\frac{dy}{dx}$: derivative of $y$ with respect to $x$;
$\frac{d}{dx}(f(x))$: operator form meaning “differentiate $f(x)$ with respect to $x$.”

If $y=x^3$, then the derivative can be written as

$$\frac{dy}{dx}=3x^2$$

$$y'=3x^2$$

The expression being differentiated is called the function or original function. The result is the derived function or gradient function.

A key term is differentiable. A function is differentiable at a point if its derivative exists there. In many cases, smooth curves are differentiable, but sharp corners are not. For example, the graph of $y=|x|$ has a corner at $x=0$, so the derivative does not exist at that point.

Basic rules for finding derivatives

The limit definition explains where derivatives come from, but in practice we usually use derivative rules to work faster. Some key rules at this level include:

Power rule

If $f(x)=x^n$, then

$$f'(x)=nx^{n-1}$$

Examples:

if $f(x)=x^5$, then $f'(x)=5x^4$;
if $f(x)=x^{-2}$, then $f'(x)=-2x^{-3}$.

Constant rule

If $f(x)=c$, where $c$ is a constant, then

$$f'(x)=0$$

This makes sense because a constant function does not change.

Constant multiple rule

If $f(x)=c\,g(x)$, then

$$f'(x)=c\,g'(x)$$

Sum and difference rule

If $f(x)=u(x)\pm v(x)$, then

$$f'(x)=u'(x)\pm v'(x)$$

For example, if $f(x)=x^3-4x+7$, then

$$f'(x)=3x^2-4$$

These rules let us find derivatives much more efficiently than using the limit definition every time.

A worked example in context

Suppose the height of a plant after $t$ weeks is modeled by

$$h(t)=2t^2+3t+10$$

The derivative is

$$h'(t)=4t+3$$

This tells us the growth rate of the plant. At $t=0$, the growth rate is

$$h'(0)=3$$

At $t=4$, the growth rate is

$$h'(4)=19$$

So the plant is growing faster later on than at the beginning. This is a simple example of how derivatives describe change in a realistic setting 🌱.

If we want to estimate the growth rate around $t=2$, we can calculate

$$h'(2)=11$$

This means the plant’s height is increasing at about $11$ units per week at that moment.

Why derivatives matter in calculus

Derivatives are one half of the larger calculus picture. The other major half is integration. Differentiation studies how quantities change, while integration studies how quantities accumulate. These ideas are connected by the Fundamental Theorem of Calculus, which shows that differentiation and integration are linked.

In the IB course, derivatives are used in many later topics:

finding gradients and tangent lines;
analyzing motion in kinematics, where velocity is the derivative of displacement and acceleration is the derivative of velocity;
determining increasing and decreasing intervals;
finding stationary points and optimization problems;
modeling rates in science and economics.

For example, if displacement is given by $s(t)$, then velocity is

$$v(t)=\frac{ds}{dt}$$

and acceleration is

$$a(t)=\frac{dv}{dt}=\frac{d^2s}{dt^2}$$

This chain of ideas shows why the introduction to derivatives is such an important foundation.

Conclusion

students, the main idea of a derivative is simple but very powerful: it measures how a function changes at a single point. You can think of it as the slope of a tangent line or as an instantaneous rate of change. The formal definition uses a limit, and practical calculations use derivative rules such as the power rule and sum rule. In IB Mathematics Analysis and Approaches SL, derivatives are essential for graph behavior, kinematics, optimization, and the later study of integration. Understanding this introduction makes the rest of calculus much easier to follow.

Study Notes

A derivative measures instantaneous rate of change and the gradient of a tangent.
The formal definition is $f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$.
The average rate of change is $\frac{f(b)-f(a)}{b-a}$.
If $f'(x)>0$, the function is increasing; if $f'(x)<0$, it is decreasing; if $f'(x)=0$, the graph may have a turning point.
Common notation includes $f'(x)$, $\frac{dy}{dx}$, and $\frac{d}{dx}(f(x))$.
The power rule is $\frac{d}{dx}(x^n)=nx^{n-1}$.
The derivative of a constant is $0$.
Differentiation is connected to integration through calculus as a whole.
In motion problems, velocity is $\frac{ds}{dt}$ and acceleration is $\frac{d^2s}{dt^2}$.
Sharp corners are not differentiable because the derivative does not exist there.
Derivatives are used in graphs, motion, and optimization across the IB syllabus.