💡 The Big Idea:
Differentiation finds how fast something is changing at any point on a curve. You use it to find maximum and minimum values by discovering where the gradient equals zero.
What You Need to Know
The Gradient Function
Use dy/dx to find the gradient at any point. It's the derivative — the rate of change.
The Power Rule
If y = x^n, then dy/dx = nx^(n-1). This is the main rule for differentiation.
Turning Points
Set dy/dx = 0 to find where the curve has a peak or trough (maximum or minimum).
Classifying Turning Points
Use the second derivative, the graph shape, or the gradient-before-and-after test to say if it's a max or min.
Optimization Problems
Apply differentiation to real-world problems: maximize profit, minimize cost, find biggest volume, etc.
Multiple Variables
If a formula has many unknowns, find a constraint equation and substitute to reduce to one variable.
1. The Gradient Function (Derivative)
What is the gradient of a curve?
Imagine a car driving along a curved road. At each point, the road has a different slope — sometimes steep uphill, sometimes flat, sometimes downhill. The gradient is a number that tells you how steep the road is at that exact point.
For a curve described by an equation like y = x² + 3x + 5, the equation tells you the y-value for any x-value. But it doesn't directly tell you the steepness at each point. That's where the gradient function comes in.
Gradient Function (Derivative)
A formula, written as dy/dx (pronounced "dy by dx"), that gives you the gradient of the curve at any point. If you substitute in an x-value, you get the steepness at that point.
Example: If the curve is y = x² + 3x + 5, the gradient function is dy/dx = 2x + 3. At x = 2, the gradient is 2(2) + 3 = 7.
Why "dy/dx"?
dy means "tiny change in y" and dx means "tiny change in x". So dy/dx is the ratio of how much y changes to how much x changes — that's the definition of gradient (rise over run).
How differentiation works: The Power Rule
Differentiation is the algebraic process that transforms a curve equation into a gradient function. The most important rule is the power rule.
Power Rule
If y = x^n, then dy/dx = nx^(n-1)
How to use it:
- Take the power (n) and bring it in front of x
- Reduce the power by 1
Example: Differentiate y = x⁵
The power is 5. Bring it down: 5x
Reduce the power by 1: 5x^(5-1) = 5x⁴
Coefficients (Numbers in front)
If there's a number in front of x, multiply it by the power you bring down.
Example: Differentiate y = 2x⁵
Bring down the power: 5 × 2 = 10
Reduce the power: x^(5-1) = x⁴
Special Cases: Linear Terms and Constants
⚡ Linear Terms (like 2x, 5x, -3x)
If y = kx (where k is any number), then dy/dx = k — just the number, x disappears!