Calculus · StudentHub Lesson

Introduction to Derivatives

Understand the derivative as the instantaneous rate of change and the slope of a tangent line.

25 minAdvanced

What you will learn

Connect the concept of slope to rate of change
Understand the derivative as the slope of a tangent line
Interpret f'(x) as the instantaneous rate of change
Estimate a derivative graphically
Recognise the derivative as a limit

Watch the lesson

Derivative as a concept | Derivatives introduction | AP Calculus AB · Khan Academy

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Topic notes

Main Idea

The derivative of a function measures its instantaneous rate of change, which geometrically is the slope of the tangent line at a point on the curve.

Key Concepts

Average rate of change uses two points; instantaneous rate uses one point via a limit
The derivative generalises slope to curves, not just straight lines
Notation f'(x) or dy/dx represents the derivative

Definitions

Slope: rate of change between two points, rise over run
Tangent line: a line touching a curve at a single point, matching its direction
Derivative: the instantaneous rate of change of a function

Formulas

Average rate of change: (f(b)-f(a)) / (b-a)
Derivative definition: f'(x) = lim(h→0) [f(x+h)-f(x)] / h

Examples

For f(x)=x², the derivative f'(x)=2x, so at x=3 the slope is 6
Average speed vs instantaneous speed illustrates average vs instantaneous rate of change

Common Mistakes

Confusing average rate of change with instantaneous rate
Thinking the derivative is just a number, not a function
Forgetting the derivative differs at different points on a curve

Key concepts

Instantaneous vs average rate of changeSlope of a tangent lineDerivative notation f'(x)Limit definition of the derivativeThe derivative is a function

Important terms

Derivative: A function describing the instantaneous rate of change of another function.
Tangent line: A line touching a curve at one point, sharing its slope there.
Limit: The value a function approaches as the input approaches some value.

Worked examples

Problem

Find the derivative of f(x) = x² at x = 3

1. Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
2. f'(x) = 2x
3. Substitute x=3: f'(3)=6

Answer: 6

Problem

Describe what f'(x) represents graphically

1. Consider the graph of f(x)
2. The tangent line at a point has a slope
3. f'(x) gives that slope at each x

Answer: The slope of the tangent line to f(x) at x

Quick revision

Derivative = instantaneous rate of change
Geometrically it is the tangent line slope
Average rate uses two points; the derivative uses a limit
Notation: f'(x) or dy/dx
The derivative is itself a function
Different x-values give different derivative values

Check your understanding

Question 1 · Multiple choice

The derivative of a function at a point represents:

Question 2 · Multiple choice

What does f'(x) commonly denote?

Question 3 · Short answer

If f(x) = x², what is f'(x)?

Question 4 · Short answer

What concept from basic algebra does the derivative generalise?

Question 5 · True or false

The derivative can have different values at different points on the same curve.

Done with Introduction to Derivatives?

Exponents and the Laws of Indices