Calculus 1: Introduction to Differentiation and Important Examples
Differentiation is an operation in which we input a "differentiable" function, and output a new function, known as the derivative of the original function. This new function, evaluated at a point, tells us the instantaneous rate of change of the original function, evaluated at the same point. In this video, we go over some need to know examples of functions and their derivatives.
Calculus 2: Differentiation - Linearity and The Chain Rule
In this video, we go over two important properties of the derivative operator. These are linearity, which lets us split a differentiation problem over scaling and summing, and the chain rule, which tells us how to differentiate compositions of functions. Using these, and the basic examples from the last video, we can now differentiate a much wider range of functions.
Calculus 3: Differentiation - Product rule and quotient rule
In this video, we go over the product rule and the quotient rule, which tell us how to differentiate functions which are products of two functions, and fraction of two functions respectively. While these rules are a little more complicated that the previously covered ones, we can now differentiate any functions that is "made up" of elementary functions via repeated applications of all the rules we have covered so far.
Calculus 4: Differentiation - Finding Critical Points
In this video, we discuss critical points, and how to find them. Critical points are important points associated to a function that satisfy certain geometric properties, such as local minimums and maximums. To find these, we differentiate a function, and find its zeroes, which correspond with the desired critical points. We also discuss some simple tests we can do to classify the properties of a critical point.
Calculus 5: Differentiation From First Principles
Differentiation from First Principles is a method in which one derives the derivative directly from the formal definition of the derivative. This is the primary method for deriving some of the derivatives that we stated without proof at the start of the series. While this method is generally not used in practice, it may be needed when trying to differentiate an unfamiliar function.
Calculus 6: Introduction to Integration
Integration is an operation in which we input an "integrable" function, and return the total area underneath the graph between two fixed points. This operator can be conceptualised as the inverse operation of the derivative, so we can use out knowledge of the derivative to know some simple integrations. In general however, integration is a lot trickier that differentiation, as the rules we have for establishing new integrals are a lot more restrictive.
Calculus 7: Rules Of Integration
Integration has some simple properties that we can use to reduce complicated integrals into several smaller ones. In this video we demonstrate some of these rules, which lets us solve more complicated integrals.
Calculus 8: Integration By Substitution
Integration by substitution is a special property of integration that we can utilise to simplify integration. The idea, roughly speaking, is to substitute a complicated expression with a single variable, in the aims of reducing the integral to something solvable. The rules for doing so are discussed in this video.
Calculus 9: Integration By Parts
Integration by parts is a property of integration that allows us to change an integral of a product, assuming we know an anti-derivative of one functions in the product. Notice that this method results in an integral at the end, so we must be careful when applying it in order to ensure we get something solvable at the end.