Derivative

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Derivative: meaning, method and solved example (AP Calculus AB) – Maths PDF

In short: Derivative measures how quickly a function’s output is moving at one exact input, and on the graph it shows up as the steepness of the line that grazes the curve there.

AP Calculus ABCalculusAP Calculus AB Unit 2

What is Derivative?

A derivative answers the question “how fast, right now?” rather than “how fast, on average?”. Think of a speedometer needle: it reports your speed this very second, not for the whole trip.

For a function \(f\), the number \(f^{\prime}(a)\) is what the average rates \(\frac{f(a+h) – f(a)}{h}\) settle toward when \(h\) shrinks to 0, provided they settle at all. Repeating this at every input produces a new function, written \(f^{\prime}(x)\) or \(\frac{dy}{dx}\).

How to spot it

  • A question asks for velocity, growth rate or marginal cost at a specific instant instead of an average.
  • You are told to find the slope of the tangent line at a given point of a curve.
  • The notation f'(x), dy/dx or y’ appears in the problem or the answer choices.
  • A graph is curved and you must say whether the output is rising or falling at one input.

The method

  1. Identify the function and the input \(x = a\) where the rate is wanted.
  2. Differentiate: use the limit definition or a rule to get \(f^{\prime}(x)\).
  3. Substitute \(x = a\) into \(f^{\prime}(x)\) to get a single number.
  4. Interpret it with units: output units per input unit, and say whether the sign means rising or falling.
(1.5, 2.25)tangent lineslope 3y = x^21.5
At x = 1.5 the curve y = x^2 has derivative 2(1.5) = 3, the slope of the red tangent line.

Derivative: solved example

Problem

A robotics club in Ohio launches a model rocket whose height is \(h(t) = 80t – 16t^2\) feet, where \(t\) is in seconds. How fast is the rocket moving at \(t = 1.5\) seconds?

Solution

Differentiate term by term: \(h^{\prime}(t) = 80 – 32t\).

Substitute: \(h^{\prime}(1.5) = 80 – 32(1.5) = 80 – 48 = 32\). The rocket is still climbing at 32 feet per second. The positive sign confirms it is going up.

Try it

Question 1

If \(f(x) = x^3 – 4x\), find \(f^{\prime}(2)\).

Show the answer

\(f^{\prime}(x) = 3x^2 – 4\), so \(f^{\prime}(2) = 3(4) – 4 = 8\).

Question 2

The temperature in a greenhouse satisfies \(T^{\prime}(3) = -2.5\) in degrees Fahrenheit per hour. What does that say at hour 3?

Show the answer

Right at hour 3 the air is cooling by 2.5 degrees per hour; the negative sign means the greenhouse is getting colder.

Derivative vs. Rate of Change

An average rate of change compares two inputs, like miles driven divided by hours over a whole trip. The derivative squeezes those two inputs together until they meet, so it describes a single instant. For a straight line the two always agree; for a curve they usually differ.

Rate of Change

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