Introduction to Further Differentiation

In your previous studies, you learned how to differentiate basic polynomials like \(x^2\) or \(5x^3\). But the world isn’t just made of polynomials! To model things like the growth of a population, the vibrations of a guitar string, or the way a virus spreads, we need to differentiate more complex functions like \(e^x\), \(\ln x\), and trigonometric functions. This chapter introduces the "tools of the trade" for handling these complex functions and explores the second derivative, which helps us understand the "curviness" of a graph.

Note: This chapter builds on Topic 5.3 (Power Rule). If you need a refresher on the basic derivative of \(ax^n\), quickly check that chapter first!

1. Derivatives of Transcendental Functions

To succeed in Higher Level (HL) Calculus, you need to memorize a few "standard" derivatives. Think of these as the basic building blocks for more complex problems.

Trigonometric Functions

When working with angles in calculus, we always use radians.
\( \frac{d}{dx}(\sin x) = \cos x \)
\( \frac{d}{dx}(\cos x) = -\sin x \) (Watch out for that negative sign!)
\( \frac{d}{dx}(\tan x) = \frac{1}{\cos^2 x} \)

Exponential and Logarithmic Functions

\( \frac{d}{dx}(e^x) = e^x \) (The easiest one! The function is its own derivative.)
\( \frac{d}{dx}(\ln x) = \frac{1}{x} \)

Rational Powers

In Standard Level, you worked with integer exponents. In HL, the power rule \( \frac{d}{dx}(x^n) = nx^{n-1} \) works for rational numbers too (fractions and decimals).
Example: To differentiate \( \sqrt{x} \), rewrite it as \( x^{1/2} \).
\( \frac{d}{dx}(x^{1/2}) = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}} \)

Quick Tip: If you see a root (like \( \sqrt[3]{x} \)) or an \( x \) in the denominator (like \( \frac{1}{x^2} \)), always rewrite it as a power (\( x^{1/3} \) or \( x^{-2} \)) before you start differentiating!

2. The Three Big Rules

Sometimes functions are combined in messy ways. We use these three rules to "dissect" them.

A. The Chain Rule (Function inside a Function)

Use this when you have a "composite" function, like \( \sin(x^2) \) or \( (3x+1)^5 \).
The Rule: Differentiate the "outside" function while leaving the "inside" alone, then multiply by the derivative of the "inside".
\( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \)

Analogy: It’s like a Russian Nesting Doll. To get to the center, you have to open the outer layer first.

B. The Product Rule (Two functions multiplied)

Use this when you have two separate functions of \( x \) multiplied together, like \( x^2 e^x \).
The Rule: If \( y = uv \), then \( \frac{dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx} \).
Memory Aid: "First times derivative of the second, plus second times derivative of the first."

C. The Quotient Rule (Two functions divided)

Use this when one function is divided by another, like \( \frac{\ln x}{x} \).
The Rule: If \( y = \frac{u}{v} \), then \( \frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2} \).
Common Mistake: Students often swap the \( u \) and \( v \) in the numerator. Always start with the bottom function: \( v \times (\text{derivative of top}) \).

In the real world, variables are often linked. If you blow air into a balloon, the radius is increasing, and the volume is also increasing. Related rates allow us to find how fast one thing is changing if we know how fast the other is changing.

We use the Chain Rule to connect these rates. For example, if we want to find how volume changes over time (\( \frac{dV}{dt} \)) and we know how the radius changes over time (\( \frac{dr}{dt} \)):
\( \frac{dV}{dt} = \frac{dV}{dr} \times \frac{dr}{dt} \)

Step-by-Step for Related Rates:
1. Identify the rates you are given (e.g., \( \frac{dx}{dt} \)).
2. Identify the rate you need to find (e.g., \( \frac{dy}{dt} \)).
3. Find an equation linking the two variables (e.g., \( y = x^2 \)).
4. Differentiate that equation and use the Chain Rule to plug in your values.

4. The Second Derivative \( f''(x) \)

The first derivative \( f'(x) \) tells us the gradient (slope) of the graph. The second derivative \( f''(x) \) tells us how that gradient is changing. This is known as concavity.

Concavity and the "Face" Method

To remember concavity, look at the shape of the graph:
- If \( f''(x) > 0 \), the graph is concave up (it looks like a smile \(\cup\)).
- If \( f''(x) < 0 \), the graph is concave down (it looks like a frown \(\cap\)).

The Second Derivative Test

Don't worry if you find the first derivative test (using sign diagrams) tedious! The second derivative gives us a shortcut to identify stationary points (where \( f'(x) = 0 \)):
1. If \( f'(a) = 0 \) and \( f''(a) > 0 \), then \( x = a \) is a local minimum (the bottom of a smile).
2. If \( f'(a) = 0 \) and \( f''(a) < 0 \), then \( x = a \) is a local maximum (the top of a frown).
3. If \( f''(a) = 0 \), the test is inconclusive, and you must go back to the first derivative test.

Did you know? In physics, if the first derivative is velocity, the second derivative is acceleration. It’s literally the "rate of change of the rate of change"!

Summary Checklist

Key Takeaways:
- Memorize your trig and log derivatives (\( \sin x \to \cos x \), etc.).
- Use the Chain Rule for "nested" functions.
- Use the Product Rule for multiplications and Quotient Rule for divisions.
- Related rates connect different variables using time (\( t \)).
- The Second Derivative determines concavity and helps identify Max/Min points quickly.

Looking Ahead: Now that you can differentiate almost anything, you can move on to "Further integration (HL)" to learn how to reverse these processes!