Welcome to Basic Differentiation!
Hi there! Welcome to one of the most exciting parts of A Level Maths: Calculus. If you’ve ever wondered how we calculate the exact speed of a falling object at a specific split-second, or how to find the slope of a curvy roller coaster, you’re in the right place. Don't worry if it seems a bit abstract at first—once you get the hang of the patterns, it’s like learning a new superpower!
In this chapter, we are going to learn how to find the gradient (the steepness) of any curve. Unlike a straight line, which has the same steepness everywhere, a curve’s steepness changes constantly. Differentiation is simply the tool we use to measure that change.
1. The Gradient of a Curve
On a straight line, the gradient is easy to find: "rise over run." But on a curve, the gradient is different at every point. To find the gradient at a specific point, we look at the tangent.
Key Term: Tangent
A tangent is a straight line that just touches a curve at a single point without crossing through it. The gradient of the curve at that point is exactly the same as the gradient of that tangent line.
Analogy: Imagine you are driving a car along a winding mountain road. At any exact moment, if your car wheels were to point perfectly straight, the path you would fly off the road in is the tangent. Your "direction" at that specific moment is the gradient of the curve.
Quick Review Box
• The gradient of a curve changes depending on where you are on the x-axis.
• To find the gradient at a point, we find the gradient of the tangent at that point.
2. Differentiation from First Principles
How do we actually calculate the gradient of a tangent if we only have one point? We use a clever trick called First Principles. We pick a second point very close to our first point and draw a line between them (called a chord). Then, we slide that second point closer and closer until the gap between them is essentially zero.
The "h" Method:
If we have a function \( f(x) \), and we move a tiny distance \( h \) along the x-axis, our new point is \( (x + h, f(x + h)) \). The gradient between these two points is:
\( \text{Gradient} = \frac{f(x+h) - f(x)}{h} \)
As \( h \) gets smaller and smaller (approaching zero), we find the derivative. The formal notation for this is:
\( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \)
Did you know?
The symbol \( \lim_{h \to 0} \) just means "what happens to this formula as \( h \) gets so small it’s almost zero?" It’s like zooming in on a curve until it looks like a straight line!
Step-by-Step: Differentiating \( x^2 \) from First Principles
1. Start with \( f(x) = x^2 \).
2. Find \( f(x+h) \): \( (x+h)^2 = x^2 + 2xh + h^2 \).
3. Put them into the formula: \( \frac{(x^2 + 2xh + h^2) - x^2}{h} \).
4. Simplify: \( \frac{2xh + h^2}{h} = 2x + h \).
5. Apply the limit (\( h \to 0 \)): The \( h \) disappears, leaving us with \( 2x \).
3. The Derivative and Notation
When we differentiate a function, we get a gradient function. This is a new formula that tells us the gradient at any value of \( x \).
There are two main ways to write this:
1. If your equation is \( y = \dots \), the derivative is \( \frac{dy}{dx} \) (pronounced "dee-y by dee-x").
2. If your equation is \( f(x) = \dots \), the derivative is \( f'(x) \) (pronounced "f-dash-x").
Key Takeaway:
\( \frac{dy}{dx} \) is not a fraction; it is a single symbol that means "the rate of change of \( y \) with respect to \( x \)."
4. The Power Rule: The Great Shortcut
Differentiating from first principles every time would take forever! Luckily, there is a shortcut for functions in the form \( y = kx^n \).
The Rule:
To differentiate \( x^n \):
1. Bring the power down to the front to multiply.
2. Reduce the power by 1.
Formula: If \( y = kx^n \), then \( \frac{dy}{dx} = nkx^{n-1} \)
Memory Aid: "Multiply by the old power, then subtract one from the power."
Examples:
• \( y = x^3 \rightarrow \frac{dy}{dx} = 3x^2 \)
• \( y = 5x^4 \rightarrow \frac{dy}{dx} = 20x^3 \) (because \( 4 \times 5 = 20 \))
• \( y = 7x \rightarrow \frac{dy}{dx} = 7 \) (because \( x \) is \( x^1 \), and \( x^0 = 1 \))
• \( y = 10 \rightarrow \frac{dy}{dx} = 0 \) (The gradient of a flat horizontal line is always zero!)
5. Working with Rational and Negative Powers
The power rule works for any power \( n \), including fractions and negative numbers. You just need to remember your laws of indices!
Negative Powers:
If \( y = \frac{1}{x^2} \), rewrite it as \( y = x^{-2} \) first.
Then, \( \frac{dy}{dx} = -2x^{-3} \), which is the same as \( -\frac{2}{x^3} \).
Fractional Powers (Roots):
If \( y = \sqrt{x} \), rewrite it as \( y = x^{1/2} \) first.
Then, \( \frac{dy}{dx} = \frac{1}{2}x^{-1/2} \), which is the same as \( \frac{1}{2\sqrt{x}} \).
Common Mistake to Avoid:
When differentiating negative powers, the number looks like it’s getting "bigger" (e.g., \( -2 \) becomes \( -3 \)). Remember, you are subtracting 1. On a number line, going from \( -2 \) to \( -3 \) is moving to the left, which is decreasing.
6. Differentiating Sums and Differences
If you have an expression with multiple terms, just differentiate each term one by one. It’s that simple!
Example:
If \( y = 2x^3 - 5x + 4 \)
Differentiate \( 2x^3 \) to get \( 6x^2 \)
Differentiate \( -5x \) to get \( -5 \)
Differentiate \( 4 \) to get \( 0 \)
So, \( \frac{dy}{dx} = 6x^2 - 5 \)
7. Sketching Gradient Functions
Sometimes you need to sketch a graph of the gradient (\( \frac{dy}{dx} \)) based on the graph of \( y \).
• Where the original graph is sloping up, the gradient graph is above the x-axis (positive).
• Where the original graph is sloping down, the gradient graph is below the x-axis (negative).
• Where the original graph is flat (turning points), the gradient graph crosses the x-axis (zero).
Key Takeaways for the Chapter:
• Differentiation finds the gradient of a curve at any point.
• First Principles is the formal proof using \( \lim_{h \to 0} \).
• The Power Rule is your best friend: \( \text{power} \times \text{coefficient} \), then \( \text{power} - 1 \).
• Always rewrite roots as fractional powers and fractions as negative powers before you start.
• The derivative of a constant is always zero.