Learn: Differentiation

Welcome!Today we'll explore differentiation, a key concept in calculus. It's all about rates of change and how functions behave. Let's get started!

What is Differentiation?Differentiation is a way to find the rate at which a function changes at any point. It's like finding the slope of a curve at a specific spot. This is useful for understanding things like speed, growth rates, or optimising processes.

The DerivativeThe derivative of a function represents the rate of change. For example, if you have a graph showing distance over time, the derivative tells you the speed at any point.

Basic Rules of DifferentiationThere are some simple rules to find derivatives:Constant Rule: The derivative of a constant is 0.Power Rule: For x^n, the derivative is n * x^(n-1).Sum Rule: The derivative of a sum is the sum of the derivatives.

Application: TangentsOnce we differentiate, we can calculate the slope of the tangent to a curve at a specific point. Tangents help us understand the behaviour of the function at that point.

Second DerivativeThe second derivative is the derivative of the first derivative. It tells us about the curvature of a graph. For example, it helps identify whether a function is concave up or concave down at a point.

Chain RuleThe chain rule is used when differentiating composite functions. It works by multiplying the derivative of the outer function by the derivative of the inner function.

Review Time!Great work! You've learned about differentiation, derivatives, tangents, and the chain rule. Let's test your understanding with a few questions.

Great Job!You've done an excellent job learning differentiation! Keep practising, and you'll master these concepts in no time.