Differentiation

Introduction to Differentiation

Welcome to one of the most powerful chapters in Mathematics! Differentiation is essentially the study of change. Whether it's a car accelerating, a population growing, or the price of a stock fluctuating, differentiation allows us to calculate exactly how fast that change is happening at any given moment.

Don't worry if this seems tricky at first! While the symbols might look new, the core idea is something you already know: finding the gradient (steepness) of a line. In this chapter, we just learn how to find that steepness even when the line is a curve.

1. The Derivative as a Gradient

In GCSE, you found the gradient of a straight line using "rise over run." In A-Level, we use differentiation to find the gradient of a tangent to a curve at any specific point \( (x, y) \).

We use the notation \(\frac{dy}{dx}\) or \(f'(x)\) to represent the first derivative. This tells us the rate of change of \(y\) with respect to \(x\).

Differentiation from First Principles

This is the "proof" of where differentiation comes from. Imagine two points on a curve that are very close together. As the distance between them (\(h\)) gets closer and closer to zero, we find the exact gradient at a single point.

The formula you need to know is:
\( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \)

Quick Review: To prove the derivative of \(x^2\), substitute \(f(x) = x^2\) into the formula, expand, and watch as the \(h\) terms simplify!

Key Takeaway:

The derivative \( \frac{dy}{dx} \) is just a formula that gives you the gradient of the curve at any point.

2. The "Toolkit" of Basic Rules

You don't always have to use "First Principles." We have shortcuts for common functions!

Powers of x

If \( y = ax^n \), then \( \frac{dy}{dx} = anx^{n-1} \).
Memory Trick: "Bring the power down to the front (multiply), then subtract one from the power."

Exponential and Log Functions

• If \( y = e^{kx} \), then \( \frac{dy}{dx} = ke^{kx} \). (The "e-asy" function stays the same, just multiply by the coefficient of \(x\)).
• If \( y = a^{kx} \), then \( \frac{dy}{dx} = ka^{kx} \ln a \).
• If \( y = \ln x \), then \( \frac{dy}{dx} = \frac{1}{x} \).

Trigonometric Functions

Angles must be in radians for these to work!

• If \( y = \sin kx \), then \( \frac{dy}{dx} = k \cos kx \)
• If \( y = \cos kx \), then \( \frac{dy}{dx} = -k \sin kx \)
• If \( y = \tan kx \), then \( \frac{dy}{dx} = k \sec^2 kx \)

Common Mistake: Forgetting the minus sign when differentiating cos! Remember: "Differentiation of Co-functions results in a Negative."

Key Takeaway:

Memorize the standard derivatives for trig, ln, and \(e^x\). They are the building blocks for everything else.

3. Stationary Points and Curve Sketching

A stationary point occurs whenever the gradient is zero (\( \frac{dy}{dx} = 0 \)). This is like the top of a hill or the bottom of a valley where, for a split second, you aren't going up or down.

Types of Stationary Points:

• Local Maximum: The peak of a hill.
• Local Minimum: The bottom of a valley.
• Stationary Point of Inflection: A "shelf" where the curve flattens but then continues in its original direction.

The Second Derivative Test

We use \( \frac{d^2y}{dx^2} \) (the derivative of the derivative) to see what kind of point we have:

• If \( \frac{d^2y}{dx^2} > 0 \), it's a Minimum (think "positive people smile," and a smile is a minimum curve).
• If \( \frac{d^2y}{dx^2} < 0 \), it's a Maximum (think "negative people frown," and a frown is a maximum curve).
• If \( \frac{d^2y}{dx^2} = 0 \), the test is inconclusive; you must check the gradient on either side of the point.

Did you know?

A Point of Inflection is specifically where the curve changes from being "convex" (bulging outwards) to "concave" (hollowed inwards). This happens when \( \frac{d^2y}{dx^2} \) changes sign.

4. The Chain, Product, and Quotient Rules

Sometimes functions are "nested" inside each other or multiplied together. We need special rules for these.

The Chain Rule (The Onion Rule)

Used for "functions of functions," like \( y = (3x + 2)^5 \).
\( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \)
Analogy: Think of it like peeling an onion. Differentiate the outside layer, then multiply by the derivative of the inside layer.

The Product Rule

Used when two functions of \(x\) are multiplied, e.g., \( y = x^2 \sin x \).
If \( y = uv \), then \( \frac{dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx} \).

The Quotient Rule

Used when one function is divided by another, e.g., \( y = \frac{e^x}{x^2} \).
If \( y = \frac{u}{v} \), then \( \frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2} \).
Memory Aid: "Low D-High minus High D-Low, over Low-Low" (where D means differentiate).

Key Takeaway:

Always identify the structure of the equation first. Is it a product? A fraction? A bracket? This tells you which rule to use.

5. Tangents and Normals

A tangent is a straight line that just touches a curve at a point. A normal is a straight line perpendicular to the tangent at that same point.

Step-by-Step Process:
1. Differentiate the curve to find \( \frac{dy}{dx} \).
2. Substitute the \(x\)-coordinate to find the gradient of the tangent (\(m_1\)).
3. For the normal, the gradient is \(m_2 = -\frac{1}{m_1} \).
4. Use the equation of a straight line: \( y - y_1 = m(x - x_1) \).

6. Implicit and Parametric Differentiation

Implicit Differentiation

Sometimes \(x\) and \(y\) are mixed together and you can't get \(y\) on its own (e.g., \( x^2 + y^2 = 25 \)).
When you differentiate a term with \(y\), treat it like \(x\) but multiply by \( \frac{dy}{dx} \).
Example: Differentiating \( y^2 \) with respect to \(x\) gives \( 2y \frac{dy}{dx} \).

Parametric Differentiation

Sometimes \(x\) and \(y\) are both defined by a third variable, usually \(t\) or \( \theta \) (e.g., \( x = 2t, y = t^2 \)).
To find the gradient:
\( \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \)

Quick Review Box:

• Increasing Function: \( \frac{dy}{dx} \geq 0 \) for the whole interval.
• Decreasing Function: \( \frac{dy}{dx} \leq 0 \) for the whole interval.
• Perpendicular Gradients: Multiply to give \(-1\).

7. Connected Rates of Change

This is where differentiation meets the real world! If you know how fast a radius is growing (\( \frac{dr}{dt} \)), you can find how fast the area of the circle is growing (\( \frac{dA}{dt} \)).

We use the chain rule to link them:
\( \frac{dA}{dt} = \frac{dA}{dr} \times \frac{dr}{dt} \)

Key Takeaway:

Always write down what you know (e.g., \( \frac{dr}{dt} = 2 \)) and what you want to find (e.g., \( \frac{dV}{dt} \)). Then find a formula that connects the variables (like \( V = \frac{4}{3}\pi r^3 \)) and differentiate it.