Learn: Differentiation

Welcome!Today we'll explore differentiation, a key concept in calculus. You'll learn how to find the rate of change for functions and its practical applications. Let's dive in!

What is Differentiation?Differentiation is the process of finding the derivative of a function, which tells us how the function changes at any given point. It helps us understand rates of change, like speed or gradient.

Why is Differentiation Useful?It allows us to measure how quickly something changes. For example, the derivative of a position function gives velocity, while the derivative of velocity gives acceleration.

The DerivativeThe derivative of a function is its rate of change. If the function is f(x), its derivative is written as f'(x) or \frac{df}{dx}. It shows how f(x) changes with respect to x.

Basic Rules of DifferentiationHere are some important rules:Power Rule: If f(x) = x^n, then f'(x) = nx^(n-1).Constant Rule: The derivative of a constant is 0.Sum Rule: The derivative of f(x) + g(x) is f'(x) + g'(x).

Applications of DifferentiationWe use differentiation to find:Gradients: To determine the slope of a curve at a point.Stationary Points: To find where a function has maximum or minimum values.Velocity and Acceleration: In physics, derivatives measure how position changes over time.

Stationary PointsA stationary point occurs when the derivative is 0, meaning the slope of the curve is flat. These points can be maxima, minima, or points of inflection.

Review Time!Fantastic work! You've learned the basics of differentiation, including its rules and applications. Let's test your understanding with a few more questions.