Learn: Differentiation

Welcome!Today we will explore differentiation, a key concept in calculus that helps us understand rates of change. This lesson is tailored for you to make learning fun and engaging!

What is Differentiation?Differentiation is a method to find the rate at which one quantity changes with respect to another. It is used to calculate gradients of curves or instantaneous rates of change.

Why is Differentiation Important?It helps us understand how things change over time, such as speed increasing or decreasing. For example, if you know the position of a car at different times, differentiation helps find its speed at any specific moment.

Basic Rules of DifferentiationThere are standard rules for differentiation. For example, the power rule states: for f(x) = x^n, the derivative is f'(x) = n * x^(n-1). This rule is widely used and forms the basis for many calculations.

Applications of DifferentiationOne common use is finding instantaneous velocity. If a ball is dropped, its position changes over time. Differentiation can tell us how fast it is falling at any given moment.

Chain RuleThe chain rule is used to differentiate composite functions. For example, if f(x) = g(h(x)), the derivative is obtained by multiplying the derivative of g by the derivative of h.

Review Time!Fantastic work! You've learned about differentiation, basic rules, and its applications. Let’s test your understanding with a few more questions.

Chain Rule PracticeIf f(x) = (3x + 2)^2, use the chain rule to find the derivative. Hint: First differentiate the outer function, then the inner.

Lesson Complete!Well done! You've mastered the basics of differentiation, including rules and applications. Keep practising to build confidence.