Welcome to the Second Derivative!

Hey everyone! Ready to level up your calculus skills? In this chapter, we're going to explore the second derivative. If the first derivative tells us how fast something is changing (like speed), the second derivative tells us how fast that *change* is changing (like acceleration!).

Don't worry if that sounds a bit meta. We'll break it down with simple examples. Understanding the second derivative is a super-power that lets you understand the true shape of a curve, easily find maximum and minimum points, and much more. Let's get started!


What is the Second Derivative? (The Basics)

Quick Review: The First Derivative

Before we jump into the second derivative, let's remember what the first derivative does.

  • For a function y = f(x), the first derivative is written as f'(x) or $$ \frac{dy}{dx} $$.
  • It tells us the rate of change of the function at any point.
  • Geometrically, it gives us the slope of the tangent to the curve at that point.

Introducing the Second Derivative

The idea is actually very simple: The second derivative is just the derivative of the first derivative. You just differentiate twice!

If the first derivative, f'(x), tells you the slope of the curve, then the second derivative tells you how the slope is changing. Is the slope getting steeper? Flatter? This information reveals the curve's shape.

Notations You Must Know

Just like the first derivative, there are a few ways to write the second derivative. The HKDSE syllabus requires you to recognise all of them:

  • If you start with f(x), the second derivative is f"(x) (we say "f double prime of x").
  • If you start with y, the second derivative is y" (we say "y double prime").
  • The Leibniz notation is $$ \frac{d^2y}{dx^2} $$ (we say "d squared y by dx squared").

How to Calculate the Second Derivative

This is the easy part! It's a simple two-step process.

Step 1: Find the first derivative of the function, f'(x).
Step 2: Differentiate your result from Step 1. That's it!

Example Walkthrough

Let's find the second derivative of f(x) = 4x³ - 5x² + 7x - 10.

  1. Find the first derivative, f'(x).
    Using the power rule, we get:
    f'(x) = 12x² - 10x + 7

  2. Differentiate f'(x) to get f"(x).
    Now we differentiate our new function:
    f"(x) = 24x - 10

And that's our answer! The second derivative of 4x³ - 5x² + 7x - 10 is 24x - 10.

Key Takeaway

The second derivative, written as f"(x) or $$ \frac{d^2y}{dx^2} $$, is found by differentiating a function twice. It measures the rate of change of the slope.


The Story of Concavity

What is Concavity?

Concavity describes the way a curve bends. Think of it like a bowl.

  • Concave Up: The curve opens upwards, like a cup or a smiley face 🙂. It can "hold water".
  • Concave Down: The curve opens downwards, like a cap or a frowny face 😟. It would "spill water".

Connecting the Second Derivative to Concavity

This is the most important application! The sign of the second derivative tells you the concavity of the graph.

  • If f"(x) > 0 on an interval, the graph is concave up on that interval.
    Memory Aid: Positive is "plus", which makes you smile 🙂.

  • If f"(x) < 0 on an interval, the graph is concave down on that interval.
    Memory Aid: Negative is "minus", which makes you frown 😟.

Points of Inflection: Where the Concavity Changes

A point of inflection is a point on the curve where the concavity changes (from up to down, or from down to up). It's the point where the curve switches its "mood"!

How to Find Points of Inflection:
  1. Find the second derivative, f"(x).
  2. Find the x-values where f"(x) = 0 (or where f"(x) is undefined). These are your potential points of inflection.
  3. Check the sign of f"(x) on either side of these x-values. If the sign (+/-) changes, then you've found a point of inflection!
Example Walkthrough

Find the point(s) of inflection for f(x) = x³ - 6x².

  1. Find f"(x).
    f'(x) = 3x² - 12x
    f"(x) = 6x - 12

  2. Solve f"(x) = 0.
    6x - 12 = 0
    6x = 12
    x = 2

  3. Test the sign of f"(x) around x = 2.
    Let's test a point to the left, like x = 1: f"(1) = 6(1) - 12 = -6 (Negative, so concave down).
    Let's test a point to the right, like x = 3: f"(3) = 6(3) - 12 = 6 (Positive, so concave up).

Since the concavity changes at x = 2, there is a point of inflection there. To get the full coordinate, find f(2) = (2)³ - 6(2)² = 8 - 24 = -16.
The point of inflection is (2, -16).

Key Takeaway

The sign of f"(x) tells you the shape: f"(x) > 0 means concave up (🙂) and f"(x) < 0 means concave down (😟). A point of inflection is where this concavity changes.


The Second Derivative Test (Finding Maxima and Minima)

Quick Review: Stationary Points

Remember that a stationary point is where the slope is zero, so f'(x) = 0. These points can be a local maximum, a local minimum, or neither. The First Derivative Test helps us figure this out, but the Second Derivative Test is often a faster shortcut!

The Test Explained

The Second Derivative Test uses concavity to classify a stationary point. Think about it:

  • A local minimum point looks like the bottom of a valley, which is a concave up shape.
  • A local maximum point looks like the top of a hill, which is a concave down shape.

This leads us to a simple test.

Step-by-Step: Using the Second Derivative Test
  1. Find all stationary points by solving f'(x) = 0. Let's call one solution x = c.
  2. Find the second derivative, f"(x).
  3. Substitute your stationary point c into the second derivative, finding f"(c).
  4. Check the sign of the result:
    • If f"(c) > 0 (concave up 🙂), then (c, f(c)) is a local minimum.
    • If f"(c) < 0 (concave down 😟), then (c, f(c)) is a local maximum.
    • If f"(c) = 0, the test is inconclusive. It tells us nothing! You must go back and use the First Derivative Test (checking the sign of f'(x)).

Common Mistakes to Avoid

  • Mistake 1: Applying the test to a point that isn't stationary. You MUST confirm f'(c) = 0 first!
  • Mistake 2: Assuming f"(c) = 0 means it's a point of inflection. It might be, but for the purpose of finding max/min, it just means the test failed and you need another method.
Key Takeaway

For a stationary point x = c, check the sign of f"(c). Positive > Minimum. Negative > Maximum. Zero > Test Fails.


Application in the Trapezoidal Rule

Did you know?

The second derivative can tell you if your approximation using the trapezoidal rule is too big or too small! This is a common question in the HKDSE.

Concavity and Trapezoids

Imagine drawing a curve and then drawing a trapezoid under it. The top of the trapezoid is a straight line connecting two points on the curve.

  • If the curve is concave up (like a bowl), the straight line will be above the curve. This means the area of the trapezoid is bigger than the true area under the curve.
  • If the curve is concave down (like a cap), the straight line will be below the curve. This means the area of the trapezoid is smaller than the true area under the curve.

The Rule for Over/Underestimation

This gives us a very simple rule for the interval you are integrating over:

  • If f"(x) > 0 (concave up) for the entire interval, the trapezoidal rule gives an overestimate.
  • If f"(x) < 0 (concave down) for the entire interval, the trapezoidal rule gives an underestimate.
Step-by-Step Guide

Example: Is the trapezoidal rule estimate for $$ \int_1^3 \frac{1}{x} dx $$ an overestimate or an underestimate?

  1. Let f(x) = 1/x = x⁻¹.

  2. Find the second derivative, f"(x).
    f'(x) = -1x⁻² = -1/x²
    f"(x) = (-1)(-2)x⁻³ = 2/x³

  3. Check the sign of f"(x) on the interval [1, 3].
    For any x between 1 and 3, x is positive. Therefore, x³ is positive, and f"(x) = 2/x³ will also be positive.
    So, f"(x) > 0 on the interval.

  4. Apply the rule.
    Since the function is concave up on the interval, the trapezoidal rule will produce an overestimate.
Key Takeaway

To determine if the trapezoidal rule gives an over- or underestimate, just check the sign of f"(x) on the interval. Concave Up (f" > 0) ➔ Overestimate. Concave Down (f" < 0) ➔ Underestimate.


Chapter Summary & Checklist

Great job making it through! The second derivative adds a rich new layer to our understanding of functions and their graphs.

Here's what we've learned:
  • The second derivative is the derivative of the first derivative.
  • The sign of f"(x) determines concavity: Positive means concave up (🙂), Negative means concave down (😟).
  • A point of inflection is where concavity changes, usually where f"(x) = 0.
  • The Second Derivative Test helps classify stationary points: at x=c where f'(c)=0, if f"(c) > 0 it's a minimum, if f"(c) < 0 it's a maximum.
  • For the Trapezoidal Rule, concave up (f" > 0) leads to an overestimate, and concave down (f" < 0) leads to an underestimate.

Keep practicing, and soon using the second derivative will feel like second nature. You've got this!