Differentiation for GCSE Maths: Explanation, Examples and Questions
Basic differentiation is an essential starting point in calculus, a branch of mathematics focused on rates of change.
At GCSE level, students learn the fundamentals of differentiating polynomials, laying the groundwork for further exploration of maths at A level.
What Does ‘Differentiation’ Mean?
Differentiation is a method in calculus used to find the rate of change of a function or to calculate the gradient of a curve at a specific point.
For example, if y = x², differentiating gives dy/dx = 2x, which tells us how the value of y changes as x changes.
The result of differentiation is called the derivative, and dy/dx is the standard notation used to represent it. The derivative shows how a function changes as its input changes.
This process is essential for finding stationary points (also known as turning points, where the gradient is zero) and determining whether they are minimum or maximum points.
Differentiation can also be used to find the equations of tangents to curves.
How to Differentiate Polynomials
1. Differentiating Single Terms of the Form axⁿ
To differentiate a term like axⁿ: multiply the coefficient (a) by the power (n) and then subtract one from the power. In symbols, if y = axⁿ, then dy/dx = a × n × xⁿ⁻¹.
Examples:
- If y = 5x³, then dy/dx = 15x².
- If y = -2x⁴, then dy/dx = -8x³.
- If y = 7x, then dy/dx = 7 (as x = x¹ and n - 1 = 0).
- If y = 3, then dy/dx = 0 (constants have no rate of change).
2. Differentiating Polynomials with Multiple Terms
When differentiating a polynomial with multiple terms, you just need to differentiate each term separately. For example, if y = 4x⁵ - 3x² + 7x - 4, differentiate each term:
- The derivative of 4x⁵ is 20x⁴.
- The derivative of -3x² is -6x.
- The derivative of 7x is 7.
- The derivative of -4 is 0.
Combining these results gives dy/dx = 20x⁴ - 6x + 7.
This approach works for any polynomial by treating each term individually.
This video explains how to find the gradients of curves and determine the equations of tangents and normals. Step-by-step examples are provided to ensure clarity and understanding.
Please note: Students not required to learn how to find the equations of normals for GCSE maths.
The rest of the lesson on differentiation is not quite ready but if you already know what you're doing we have practice questions linked below.