The Definition of Differentiation

Transcription

1 CALCULO DIFERENCIAL UNIVERSIDAD LIBRE FACULTAD DE INGENIEIRA DEPARTAMENTO DE CIENCIAS BASICAS LECTURAS EN INGLES The Definition of Differentiation The essence of calculus is the derivative. The derivative is the instantaneous rate of change of a function with respect to one of its variables. This is equivalent to finding the slope of the tangent line to the function at a point. Let's use the view of derivatives as tangents to motivate a geometric definition of the derivative. We want to find the slope of the tangent line to a graph at the point P. We can approximate the slope by drawing a line through the point P and another point nearby, and then finding the slope of that line, called a secant line. The slope of a line is determined using the following formula (m represents slope) : Let P = (x,y) and Q := (a,b). Let Then the slope of the line

2 Now, we chose an arbitrary interval to be Delta-x. How does the size of Delta-x affect our estimate of the slope of the tangent line? The smaller Delta-x is, the more accurate this approximation is. There is a wonderful animation of this by Douglas Arnold. Look at it here. You can see on the left of the animation how Delta-x decreases, causing the secant line the approach the tangent, where it zooms in on the right. Another animation of this (also from Douglas Arnold) is here. What we want to do is decrease the size of Delta-x as much as possible. We do this by taking the limit as Delta-x approaches zero. In the limit, assuming the limit exists, we will find the exact slope of the tangent line to the curve at the given point. This value is the derivative; There are a few different, but equivalent, versions of this definition. In more general considerations, h is often used in place of Delta-x. Or Delta-y is used in place of This leads to three commonly used ways of expressing the definition of the derivative:

3 The Notation of Differentiation Suggested Prerequesites: The definition of the derivative Often the most confusing thing for a student introduced to differentiation is the notation associated with it. Here an attempt will be made to introduce as many types of notation as possible. A derivative is always the derivative of a function with respect to a variable. When we write the definition of the derivative as we mean the derivative of the function f(x) with respect to the variable x. One type of notation for derivatives is sometimes called prime notation. The function f (x), which would be read ``f-prime of x'', means the derivative of f(x) with respect to x. If we say y = f(x), then y (read ``y-prime'') = f (x). This is even sometimes taken as far as to write things such as, for y = x 4 + 3x (for example), y = (x 4 + 3x). Higher order derivatives in prime notation are represented by increasing the number of primes. For example, the second derivative of y with respect to x would be written as

4 Beyond the second or third derivative, all those primes get messy, so often the order of the derivative is instead writen as a roman superscript in parenthesis, so that the ninth derivative of f(x) with respect to x is written as f (9) (x) or f (ix) (x). A second type of notation for derivatives is sometimes called operator notation. The operator D x is applied to a function in order to perform differentiation. Then, the derivative of f(x) = y with respect to x can be written as D x y (read ``D -- sub -- x of y'') or as D x f(x (read ``D -- sub x -- of -- f(x)''). Higher order derivatives are written by adding a superscript to D x, so that, for example the third derivative of y = (x 2 +sin(x)) with respect to x would be written as Another commonly used notation was developed by Leibnitz and is accordingly called Leibnitz notation. With this notation, if y = f(x), then the derivative of y with respect to x can be written as (his is read as ``dy -- dx'', but not ``dy minus dx'' or sometimes ``dy over dx''). Since y = f(x), we can also write This notation suggests that perhaps derivatives can be treated like fractions, which is true in limited ways in some circumstances. (For example with the chain rule.) This is also called differential notation, where dy and dx are differentials. This notation becomes very useful when dealing with differential equations. A variation of Leibnitz's differential notation is written instead as

5 which resembles the above operator notation, with (d/dx as the operator). Higher order derivatives using leibnitz notation can be written as The exponents may seem to be in strange places in the second form, but it makes sense if you look at the first form. So, those are the most commonly used notations for differentiation. It's possible that there exist other, obscure notations used by a some, but these obscure forms won't be included here. It's helpful to be familiar with the different notations. When is a function differentiable? Suggested Prerequesites: The definition of the derivative, Continuity A function is differentiable when the definition of differention can be applied in a meaningful manner to it. When would this definition not apply? It would not apply when the limit does not exist. Then, we want to look at the conditions for the limits to exist. Back to the Calculus page Back to the World Web Math top page

6 Last modified August 24, 1998 Teacher: Martha Liliana Orozco

7