Calculus · Step-by-step guide

What Is a Derivative?

Understand derivatives as slopes, rates of change, and limits through visual and numerical examples.

By Tyler Blovat··9 min read
The short answer

Start with the central idea

A derivative measures instantaneous rate of change. Geometrically it is the slope of the tangent line; numerically it is the limit of average rates over smaller intervals: f′(x)=lim⁡h→0(f(x+h)−f(x))/h\displaystyle f'(x)=\lim_{h\to0}(f(x+h)-f(x))/h.

f′(x)=lim⁡h→0f(x+h)−f(x)h\displaystyle f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}h
01

How it works

Build the method from meaning before memorizing the moves.

These are the decisions that make the procedure reliable. Read them once, then look for each idea in the worked examples below.

1

Average becomes instantaneous

A secant slope uses two points. As the second point approaches the first, the limiting secant slope becomes the tangent slope.

2

Units are output per input

Position differentiated with respect to time gives velocity; cost differentiated by quantity gives marginal cost.

3

The derivative is a function

f′(x) reports the slope of f at each x where that rate exists.

02

See the structure

A picture makes the relationships easier to remember.

tangent line
The derivative at the marked point is the slope of the highlighted tangent line.
03

Worked examples

Follow the reason for each line, then try to reproduce it without looking.

Example 1Power function

Find the slope of f(x)=x2\displaystyle f(x)=x^2 at x = 3.

  1. The derivative is f′(x)=2x\displaystyle f'(x)=2x.
  2. Evaluate f′(3)=2(3)\displaystyle f'(3)=2(3).
  3. The tangent rises 6 units per unit right.
Answer6
Example 2Constant rate

s(t)=50t+10\displaystyle s(t)=50t+10 models position in miles.

  1. The slope of this linear function is 50.
  2. s′(t)=50\displaystyle s'(t)=50.
  3. The units are miles per hour if t is hours.
Answer50 mph
Example 3Horizontal tangent

f(x)=x2−4x\displaystyle f(x)=x^2-4x at x = 2.

  1. f′(x)=2x−4\displaystyle f'(x)=2x-4.
  2. f′(2)=0\displaystyle f'(2)=0.
  3. The graph has a horizontal tangent there.
AnswerSlope 0
04

Common mistakes—and how to avoid them

Accuracy improves fastest when you know what to check.

  • Treating a derivative as the function’s y-value.

  • Forgetting rate units.

  • Assuming every sharp corner or discontinuity has a derivative.

05

Turn the explanation into a skill

Reading creates recognition. Independent practice creates recall.

Free printable worksheet · Answer key included

Derivative Rules

Practice interpreting and calculating derivatives.

Preview the worksheet

A strong practice loop: solve one example with the guide open, solve a similar problem without it, explain the method aloud, and return the next day for a short mixed review.

06

Practice problems with worked solutions

Solve each problem on paper, use the hint only if needed, then compare every step.

1

Find the average rate of change of f(x)=x2\displaystyle f(x)=x^2 from x = 1 to x = 4.

Show hint

Use the secant slope.

Show worked solution
  1. f(4)=16\displaystyle f(4)=16 and f(1)=1\displaystyle f(1)=1.
  2. (16−1)/(4−1)=15/3\displaystyle (16-1)/(4-1)=15/3.

Answer: 5\displaystyle 5

2

Use the limit definition to find the derivative of f(x)=x2\displaystyle f(x)=x^2.

Show hint

Expand (x+h)2\displaystyle (x+h)^2.

Show worked solution
  1. (f(x+h)−f(x))/h=((x+h)2−x2)/h\displaystyle (f(x+h)-f(x))/h=((x+h)^2-x^2)/h.
  2. Simplify to (2xh+h2)/h=2x+h\displaystyle (2xh+h^2)/h=2x+h.
  3. Let h→0\displaystyle h\to0.

Answer: f′(x)=2x\displaystyle f'(x)=2x

3

If position is s(t)=t2+3t\displaystyle s(t)=t^2+3t, find velocity at t = 2.

Show hint

Velocity is the derivative of position.

Show worked solution
  1. s′(t)=2t+3\displaystyle s'(t)=2t+3.
  2. s′(2)=4+3\displaystyle s'(2)=4+3.

Answer: 7 units per time.

4

What does f′(3)=−4\displaystyle f'(3)=-4 mean graphically?

Show hint

A derivative is tangent slope.

Show worked solution
  1. At x = 3, the tangent line has slope −4.
  2. The graph is decreasing there at 4 vertical units per horizontal unit.

Answer: The tangent slope at x = 3 is −4.

5

Estimate f′(2)\displaystyle f'(2) if nearby secant slopes are 3.9, 3.99, 4.01, and 4.1.

Show hint

Look for the limiting value.

Show worked solution
  1. The slopes from both sides cluster around 4.
  2. Their limiting value estimates the tangent slope.

Answer: Approximately 4.

07

Frequently asked questions

Quick answers before you move on.

What should I remember first?

A derivative measures instantaneous rate of change. Geometrically it is the slope of the tangent line; numerically it is the limit of average rates over smaller intervals: f′(x)=lim⁡h→0(f(x+h)−f(x))/h\displaystyle f'(x)=\lim_{h\to0}(f(x+h)-f(x))/h.

How do I know whether I understand this topic?

You should be able to name the method, explain why each step is valid, complete a new example without copying, and check whether your answer is reasonable.

What should I do if I keep making the same mistake?

Write the error as a specific checkpoint—for example, “I will identify the hypotenuse before substituting.” Then use that checkpoint on three short problems rather than repeating a full page without feedback.

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