Learn: Differentiation

Welcome!Today we'll explore differentiation, a key topic in AQA A Level Mathematics. It's all about finding rates of change and slopes of curves. Let's break it down step by step!

What is Differentiation?Differentiation is a method used to calculate the rate of change of one quantity with respect to another. For example, it helps us find the slope of a curve at any given point, or the speed of a moving object when its distance changes over time.

Why is Differentiation Important?Knowing how quantities change is crucial in many fields, such as physics, economics, and engineering. For example, in physics, differentiation helps calculate acceleration when speed changes over time.

Key Concepts in DifferentiationLet's review some important terms:

Gradient — The slope or steepness of a curve at a specific point.Derivative — The result of differentiation, showing the rate of change of a function.dy/dx — A common notation for the derivative, representing how y changes as x changes.

How to Differentiate Basic FunctionsTo differentiate a function, follow these basic rules:

Power Rule — Multiply the power by the coefficient and decrease the power by 1. For example, if f(x) = xn, then f'(x) = nxn-1.Constant Rule — The derivative of a constant is always 0. For example, if f(x) = 5, then f'(x) = 0.Sum Rule — Differentiate each term separately. For example, if f(x) = x2 + x, then f'(x) = 2x + 1.

Worked ExampleIf f(x) = 3x2 + 4x + 5, find f'(x).

Second DerivativeThe second derivative is the derivative of the derivative. It shows how the rate of change itself changes. For example, it can represent acceleration if the first derivative is speed.

Review Time!Great work! You've learned about differentiation, its importance, basic rules, and the second derivative. Now let's test your understanding with a few final questions.