Differentiation

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Estimate gradients of curves by drawing tangents
  2. Use the derivatives of functions of the form ax^n, where a is a rational constant and n is a positive integer or zero, and simple sums of not more than three of these
  3. Apply differentiation to gradients and stationary points (turning points)
  4. Discriminate between maxima and minima by any method
  5. Use dy/dx notation; identify maximum and minimum points by sketching, using the second differential, or inspecting the gradient either side of a turning point

1. Finding Gradients of Curves (Using Tangents)

https://cambridgeassistant.com

What is the gradient of a curve?

Unlike a straight line, which has the same gradient everywhere, a curve has a gradient that changes as you move along it. The gradient at any point on a curve tells you how steep the curve is at that exact spot.

To find the gradient of a curve at a specific point, you need to draw a tangent at that point.

Tangent: A straight line that just touches the curve at one point and points in the same direction as the curve at that point.

How to estimate the gradient using a tangent

Step 1: Draw the tangent line carefully at the point where you want to find the gradient. Make sure it only touches the curve at that one point and goes in the same direction as the curve.

Step 2: Draw a long tangent. The longer your tangent line, the more accurate your answer will be.

Step 3: Choose two points on the tangent line that are far apart from each other. Call them (x₁, y₁) and (x₂, y₂). Do not use the point where the tangent touches the curve - use two other points on the tangent instead.

Step 4: Calculate the gradient using the formula:

Gradient = (y₂ - y₁) / (x₂ - x₁)

This is the change in y divided by the change in x.

Step 5: The gradient of the tangent is equal to the gradient of the curve at that point.

What does the gradient represent?

The gradient shows the rate of change - how fast one quantity is changing compared to another.

For example:

  • On a distance-time graph, the gradient represents speed (how distance changes per unit of time)
  • On a temperature-time graph, the gradient represents how quickly temperature is changing per unit of time

Template to remember: The gradient is the rate of change of [y-axis quantity] per unit change in [x-axis quantity].

Sign in to view full notes