Home › Math Adventures › Basic_caculus› Derivatives
← Back to Math Adventures

Derivatives

Overview

Instant speed.

What is it?

The instantaneous rate of change of a function; the slope of the tangent line at a point.

History

Calculated by Newton to predict exactly where planets would be in their orbits.

Key Idea

Power Rule: Multiply by the power, then lower the power by 1.

Practice This Topic

Concept Guide

Plain English: The derivative tells you how fast something is changing right now. If the graph is 'Position', the derivative is 'Speed'. If the graph is 'Speed', the derivative is 'Acceleration'.

Real-world example: Roller Coasters. Engineers use derivatives to calculate the g-forces at every curve to ensure passengers don't black out.

How to do it

  1. Identify the term (e.g., 3x^4).
  2. Bring the exponent down and multiply it by the coefficient (4 * 3 = 12).
  3. Subtract 1 from the exponent (4 - 1 = 3).
  4. Result: 12x^3.
⚠

Common Pitfall

Treating constants like variables. The derivative of a plain number (like 5) is 0, because 5 never changes.

Word Problem
"A rocket's height in meters is given by h(t) = 5t^2. How fast is it moving upwards at exactly 4 seconds?"
Reasoning: We need velocity, which is the derivative of height. Power rule on 5t^2 gives v(t) = 10t. Plug in t=4. 10 * 4 = 40 m/s.

Practice Examples

Power Rule
f(x) = x^3 f'(x) = 3x^2
Exponent 3 moves to front, becomes 2.