Functions and their derivatives The concept of a derivative Rules of differentiation

Functions and their derivatives Today we build on what we saw last week, but introduce a bit more formalism: We already know that finding a maximum or a minimum involves finding the point for which the value of the function neither increases nor falls. What we do this week is to formalise this by introducing the mathematical approach to this In particular, we will approach the idea of the slope of a function from the point of view of the concept of a derivative of a function

Functions and their derivatives The concept of a derivative Rules of differentiation

The concept of a derivative Imagine that we want to find the maximum of a particular function of x Example How do we find out at which point it has a maximum without having to use a graph? We need to introduce the concept of a derivative

The concept of a derivative We will use the following approach: We will introduce the concept of a tangent This will allow us to introduce the concept of a derivative using the graphical approach, which is more intuitive. You are on a given point on a function, and you want to calculate the slope at that point

The concept of a derivative y x Δy = y 2 – y 1 Δx = h y 1 = f(x) xx + h y 2 = f(x + h) y = f(x)

The concept of a derivative y x Problem: Different points give different slopes Which one gives the best measurement of the slope of the curve?

The concept of a derivative y x Mathematically, the measurement of the slope gets better as the points get closer. The best case occurs for an infinitesimal variation in x. The resulting line is called the tangent.

The concept of a derivative y x The tangent of a curve is the straight line that has a single contact point with the curve, and the two form a zero angle at that point.

The concept of a derivative y x The slope of the tangent is equal to the change in y following an infinitesimal variation in x. Δy Δx

The concept of a derivative y x As we saw last week, a maximum is reached when the slope of the tangent is equal to zero.

The concept of a derivative y x B A s 1 C 1 -s Slope > 0 Slope < 0 Slope = 0

The concept of a derivative With continuous functions, each curve is made up of an infinite number of points This is because points on the curve are separated by infinitely small steps (infinitesimals) There is an infinite number of corresponding tangents This is not the case for discrete curves There is only a finite number of points (no infinitesimals) What we need is a recipe for calculating the slope of a function for any given point on it Luckily, even though there are an infinite number of points, this allows us to derive such “recipes”.

Functions and their derivatives The concept of a derivative Rules of differentiation

The concept of a derivative y x Δy = y 2 – y 1 Δx = h y 1 = f(x) xx + h y 2 = f(x + h) y = f(x)

General rule: Let f be a continuous function defined at point. The derivative of the function f’(x) is the following limit of function f at point x : In other words, it is literally the calculation of the slope as the size of the step become infinitely small.  This is done using limits, but we will use specific rules that don’t require calculating this limit every time Rules of differentiation

Example k (constant)0 f(x) = 3  f ’ (x)=0 x1 f(x) = 3x  f ’ (x)=3 f(x) = 5x ²  f ’ (x)=10x

Rules of differentiation The second derivative of a function is the derivative of the derivative: This is possible because the derivative f’(x) is itself a function of x that has a slope Example :