Derivative

Calculus

definition

Also known as: rate of change, slope function

Grade 9-12

The instantaneous rate of change of a function at a single point, defined as the limit of the slope of secant lines. Derivatives describe motion, enable optimization, and quantify how any quantity changes with respect to another.

This concept is covered in depth in our complete derivatives guide, with worked examples, practice problems, and common mistakes.

Definition

The instantaneous rate of change of a function at a single point, defined as the limit of the slope of secant lines.

💡 Intuition

How fast is the output changing right now? The slope of the curve at each point.

🎯 Core Idea

The derivative transforms a function into its 'slope function'—the rate of change at every input.

Example

If position s(t) = t^2, velocity v(t) = s'(t) = 2t At t = 3, velocity = 6.

Formula

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

Notation

f'(x), \frac{dy}{dx}, \frac{df}{dx}, or Df(x) all denote the derivative of f with respect to x.

🌟 Why It Matters

Derivatives describe motion, enable optimization, and quantify how any quantity changes with respect to another.

💭 Hint When Stuck

Try sketching the tangent line at the point and estimate its slope by picking two close points on the curve.

Formal View

f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}, provided the limit exists. Equivalently, f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}.

Related Concepts

Limit Slope Differentiation Rules Chain Rule Integral

Compare With Similar Concepts

Derivative vs Slope

🚧 Common Stuck Point

Derivative of position is velocity. Derivative of velocity is acceleration.

⚠️ Common Mistakes

Confusing the power rule exponent: the derivative of x^n is nx^{n-1}, not nx^n — the exponent decreases by 1.
Treating the derivative of a product as the product of derivatives: \frac{d}{dx}[f(x)g(x)] \neq f'(x)g'(x) — you must use the product rule.
Forgetting that the derivative of a constant is 0, not 1 — constants have no rate of change.

Common Mistakes Guides

Chain Rule Product Rule

Go Deeper

Worked Examples

Step-by-step solved problems

Practice Problems

Test your understanding

Formula Explained

Notation, derivation, and common mistakes

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

Frequently Asked Questions

What is Derivative in Math?

The instantaneous rate of change of a function at a single point, defined as the limit of the slope of secant lines.

Why is Derivative important?

Derivatives describe motion, enable optimization, and quantify how any quantity changes with respect to another.

What do students usually get wrong about Derivative?

Derivative of position is velocity. Derivative of velocity is acceleration.

What should I learn before Derivative?

Before studying Derivative, you should understand: limit, slope.

Prerequisites

limit slope

Next Steps

differentiation rules chain rule integral

Cross-Subject Connections

linear functions — Derivative gives the local linear approximation of any function changing rate — Derivative quantifies how the rate of change itself varies normal distribution — Setting the derivative to zero finds the peak of a distribution

How Derivative Connects to Other Ideas

To understand derivative, you should first be comfortable with limit and slope. Once you have a solid grasp of derivative, you can move on to differentiation rules, chain rule and integral.

Want the Full Guide?

This concept is explained step by step in our complete guide:

Derivatives Explained: Rules, Interpretation, and Applications →

Learn More

Why Students Struggle with Math — In-depth guide Why Memorizing Formulas Doesn't Work — In-depth guide

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Visualization

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