Definition of the Derivative

Learning Objective

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

• Understand what a derivative represents.

• Define the derivative formally using limits.

• Interpret the derivative geometrically as a slope of the tangent line.

• Compute derivatives from first principles.

Introduction

The concept of the derivative is one of the most important ideas in calculus.
It allows us to describe how a quantity changes — for example, how fast an object moves, how quickly a function grows, or how steep a curve is at a given point.

In simple terms, the derivative measures the rate of change or the slope of a function at a specific point.

Average Rate of Change

Before defining the derivative, let’s recall the average rate of change of a function f(x) between two points x and x+h :

$$Average rate of change=\frac{f(x+h) – h}{h}$$​

This formula gives the slope of the secant line connecting the points ( x , f(x) ) and ( x+h , f(x+h) ) on the curve.

Instantaneous Rate of Change

The derivative represents the instantaneous rate of change — the slope of the tangent line at a single point.

To find it, we take the limit as h approaches 0:

$$\lim\limit_{h \to 0}\frac{f(x+h) – f(x)}{h}$$​

This limit (if it exists) gives the derivative of f at x.

Geometric Interpretation

• The secant line approximates the function between two points.

• As h→0, the second point moves closer to $x$.

• The tangent line touches the curve at only one point — its slope is the derivative.

So, the derivative $$f^\prime(x)$$ tells us how steep the curve is at x.

Example

Let’s find the derivative of $$f(x) = x^{2}$$ from the definition.

$$f^\prime(x) = \lim\limit_{h \to 0}\frac{(x+h)^{2} -x^{2}}{h}$$​

Simplify the numerator:

$$(x+h)^{2}-x^{2} = x^{2}+2xh+h^{2}-x^{2}=2xh+h^2$$

So,

$$f^\prime(x) = \lim\limit_{h \to 0}\frac{2xh + h^{2}}{h}=\lim\limit_{h\to0}(2x+h)=2x$$

✅ Therefore, $$f^\prime(x) = 2x$$.

Notations for Derivative

There are several common notations for the derivative:

$$f^\prime(x) , \frac{dy}{dx} , y^\prime$$

All of them represent the same idea — the rate at which y changes with respect to x.

Real-World Meaning

• In physics, $$\frac{dy}{dx}$$ is velocity — the rate of change of position with respect to time.

• In economics, $$\frac{dC}{dx}$$​ could represent marginal cost — how cost changes as production changes.

• In biology, it can represent growth rates of populations.

When the Derivative Does Not Exist

A function may not be differentiable at certain points if:

• It has a corner (like ∣x∣ at x=0)

• It has a cusp

• It has a discontinuity

• The tangent line is vertical

In these cases, the derivative does not exist.