EDEXCEL GCSE HIGHER MATHS - What is Differentiation and Tangents

Edexcel GCSE Higher Maths: Differentiation and Tangents

Introduction

Differentiation is a fundamental concept in calculus that allows us to find the rate of change of a function at a specific point. This rate of change is represented by the gradient of the tangent line to the curve at that point.

What is a Tangent?

A tangent line to a curve at a point is a straight line that touches the curve at that point and has the same gradient as the curve at that point. It represents the instantaneous direction of the curve at that specific point.

Differentiation

Differentiation is the process of finding the derivative of a function. The derivative of a function represents the instantaneous rate of change of the function with respect to its input variable.

Key Concepts:

• Derivative: The derivative of a function f(x) is denoted by f'(x) or df/dx.
• Gradient of the Tangent: The derivative of a function at a point x = a gives the gradient of the tangent line to the curve at that point.

Finding the Equation of a Tangent

1. Find the derivative of the function.
2. Substitute the x-coordinate of the point of contact into the derivative to find the gradient of the tangent.
3. Use the point-slope form of a line to find the equation of the tangent.

Example:

Find the equation of the tangent to the curve y = x² at the point (2, 4).

1. Find the derivative: f'(x) = 2x

2. Find the gradient at x = 2: f'(2) = 2(2) = 4

3. Use the point-slope form: y - 4 = 4(x - 2)

4. Simplify to get the equation in slope-intercept form: y = 4x - 4

Applications of Differentiation

Differentiation has numerous applications in various fields, including:

• Physics: Determining velocity and acceleration of objects
• Engineering: Designing optimal structures and systems
• Economics: Analyzing market trends and predicting economic growth
• Medicine: Modeling disease progression and optimizing treatment plans

Summary

Differentiation is a powerful tool for understanding the rate of change of functions. By finding the derivative, we can determine the gradient of the tangent line to a curve at a specific point, providing valuable insights into the behavior of the function at that point. Tangents are essential for visualizing the instantaneous direction of a curve and have wide-ranging applications in various disciplines.