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Derivatives

The Hook

Instant speed.

What is it?

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

Time Travel (History)

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

Cheat Code

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

Practice This Now!

The Field Guide

In 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'.

In The Real World: 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.
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Booby Trap!

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

Real World Challenge
"A rocket's height in meters is given by h(t) = 5t^2. How fast is it moving upwards at exactly 4 seconds?"
The Logic: 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.

Training Drills

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