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MATH 1441 Technical Mathematics for Biological Sciences The Derivative The derivative is the fundamental "thing" of differential calculus. It is nothing more than a formula for the slope of a graph. Coming up with ways of determining this formula involves the determination of a relatively straightforward limit (thought the algebra can be quite tedious). Obtaining a formula for the slope of a graph may not sound very exciting or very useful. However, slopes of graphs are really measures of rates of change. And most of the principles governing the way the world works seem to involve rates of change. Thus, mathematical machinery for working with rates of change will be fundamental tools in understanding much of physics, chemistry, and biology. The Slope of A Graph We have a clear and easy to use definition of slope for a graph that consists of a straight line: slope rise run (1) This ratio of rise/run is the same between any two points on a straight line, and so, we can easily calculate the slope of the entire line by just using the coordinates of any two distinct points on that line in formula (1). The slope measures how steep the line is: a large valued positive slope indicates a line rising steeply from left to y right. A small valued positive slope indicates a line rising only slowly from left to right. Negative slopes indicate lines C dropping as you move along them from left to right. This notion of steepness is not difficult to visualize for curved graphs. The sketch to the right shows such a graph with four locations labeled A, B, C, and D. Near A, the graph is rising slowly , near B it is rising more rapidly, near C, it is not rising or falling much, and near D is appears to be fallilng quite rapidly. It is natural to think of the slope of this graph as being positive but small near A, positive and larger near B, nearly zero near C, and negative and quite large numerically near D. A big difference from the situation with a straight line graph is that when the graph is a curve, the slope is not a single number that applies to the whole graph, but is a quantity that varies from region to region, even from point to point, along the graph, being positive in some regions, negative in others, with values ranging from small numbers or even zero at some points, to quite large numbers in other regions. D B A x y The idea behind the formula for the slope of a curved graph starts by identifying the slope of the curve at a point as equal to the slope of the so-called tangent line to the curve at that point. The tangent line is a line that just touches the curve at that precise point. Three such lines x are sketched on the graph to the right. You can see that the tangent lines seem to have the same slope as the curve at the points where they touch. The advantage of this is that we know how to calculate the slope of a line, and so to get a formula for the slope of the curve from point to point, we just need to find a formula for the slope of the sequence of tangent lines. David W. Sabo (1999) The Derivative Page 1 of 8 The next difficulty we encounter is that we need two points on a line to be able to calculate its slope, but we only know the coordinates of one point on the tangent line -- the coordinates of that single point where it touches the curve. Solving this problem involves one of the relatively few new mathematical ideas in calculus. The -Method We will illustrate the way a formula for the slope of a tangent line is obtained using a specific example, and then we will state a general formula. This general formula embodies what is usually called the -method for finding derivatives. It can be quite cumbersome, particularly for complicated formulas, so it is not used for routine work in calculus. Nevertheless, it is the starting point of all formulas for derivatives. y y = x2 As an example, consider the graph of y = x2, shown with a lot of other stuff in the figure to the right. The focus is on the lower point with x-coordinate denoted 'x'. The dashed line through that point represents the tangent line at that point, and so the goal is to determine the slope of the dashed line. The difficulty, as mentioned before, is that we know the coordinates of only one point on this line, which is insufficient information to be able to compute its slope. (x + x, y + y) y The -method uses the following stratagem. Pick a second point on the curve some distance away from the point of interest this is the point shown in the figure with coordinates (x + x, y + y). Here the symbols x and y are intended to indicate small quantities. We can now easily write down the slope of the line between these two points (at least formally), since with the notation adopted, the "rise" of this line is just y, and the "run" is just x: x x x x + x slope rise y run x In fact, since the second point is the point on the curve with horizontal coordinate x + x, we can even express y in terms of x: y x x x 2 2 So, we can express the slope of the line (shown in the figure as a solid line sometimes called a secant line for the curve) between the two points in terms of the quantity x, the horizontal distance between the point of interest at (x, y), and the arbitrarily chosen second point at (x + x, y + y): 2 rise y x x x slope run x x 2 The thing to notice is that if we picture shifting the upper point towards the lower point (that is, decrease the value of x towards zero), the solid secant line becomes more and more like the dashed tangent line we wish to determine, and in particular, the formula for the slope of the secant line given just above must give a better and better estimate of the desired slope of the tangent line the closer x gets to zero. If we could get a useful result out of this slope formula by substituting x = 0 into it, our problem would be solved, since when x = 0, the two lines would have to be identical. Unfortunately, substituting x = 0 into this formula Page 2 of 8 The Derivative David W. Sabo (1999) gives us the problematic result 0/0. This is not surprising in a way, because setting x = 0 is really the same thing as trying to calculate the slope of a line by using the coordinates of the same point twice. All is not lost however. Although we cannot set x exactly to zero, we can imagine its value decreasing to as close to zero as we wish, and so, it seems reasonable to propose the following: y x 0 x slope of the tangent line lim lim x x 2 x2 x x 0 x 2 2x x x x 2 2 lim x x 0 lim x 2x x x 0 x lim 2 x x 2 x x 0 The use of lim x 0 allows us to think of x becoming as close to zero as we wish, but not exactly zero. This avoids the problem with the 0/0 result, but still permits approximating the slope of the tangent line by the slope of a line which is for all practical purposes indistinguishable from the tangent line. The series of algebraic manipulations just above is prompted by our experience with the examples of limits in the preceding document that involved the 0/0 form: this form must arise because there is a common factor of x in the numerator and denominator of the expression. We simplify the numerator, and factor it so that the common factor can be cancelled, leaving us with evaluating the limit of 2x + x as x 0. But obviously, the closer to zero x is, the more and more this expression looks like just 2x. The conclusion is that the slope of the graph of y = x2 is given by 2x. This should seem fairly plausible. For instance, at the vertex of the parabola at x = 0, the tangent line is horizontal, and this expression gives 2 0 = 0. If we consider a sequence of increasing positive values of x, we know that the parabola is increasing more steeply to the right, and the expression 2x will give us increasing positive values. On the other hand, the left-half of the parabola has a negative slope, and for negative values of x, 2x gives a negative value. So, what we have found is that the expression that gives the slope of the graph of y = x 2 is just 2x, and so we would call this expression, 2x, the derivative of the function y = x2 with respect to x. Several different notations are used to denote the derivative: d dy [y ] y 2x dx dx We recommend the first form shown here, because it emphasizes that the determination of the derivative is an operation that is done on the function y. You can think of the symbol d/dx as similar to the symbols or log, etc., in that they stand for a specific kind of operation that is done to some expression to which they are applied. You should not regard the symbol d/dx or even dy/dx as a fraction, nor should you consider dx to indicate "d times x". Using primes to denote derivatives, as in y', is very popular because it minimizes the amount of writing, but it can be confusing, since no mention is made in this notation of what is the intended independent variable. In the present example, the only other variable is x, and so it is reasonable to interpret the symbol y' as meaning the derivative of y with respect to x. However, when a calculation involves several variables, the exact meaning of y' may be ambiguous. David W. Sabo (1999) The Derivative Page 3 of 8 You need to study the example above very carefully to make sure you understand the mathematical and geometric strategies it involves. The limiting process used to get the slope of the tangent line was initially quite controversial, but mathematicians now understand the circumstances under which it is justified, and so use this approach without reservation for defining derivatives as required in typical scientific and technical applications. We can now state the gist of the -method in some generality. For the graph of the function y = f(x), the derivative, or the formula for the slope of the graph when y is plotted against x, is given by f x x f x d y lim x 0 dx x (2) To implement this formula, you simply use the function definition f(x) to get the initial expression for the quotient whose limit is required, manipulate the numerator until a x can be factored out in order to cancel the x in the denominator, and then usually, the value of the desired limit will be obvious by inspection. The -method is the fundamental formula of differential calculus. However, it is not used routinely to find derivatives. Instead, what we will do in the next few documents in this series is use the -method formally to derive recipes that relate the derivative of an expression to derivatives of its simpler parts. By applying these recipes, the derivatives of even very complicated expressions can be obtained relatively easily without going through all the work of the -method from scratch. However, even though you will not use the -method routinely for determining derivatives, you do need to understand the principles of the method, and how to use it for very simple expressions. To help you with this, we conclude this document with a few more quick examples of the method. Example 1: Use the -method to determine the derivative of y = x3 with respect to x. Solution From the basic definition x x x 3 d d 3 y dx x lim x 0 dx x 3 x 3 3 x 2 x 3 x x x x 3 2 lim x 0 3 x 2 x 3 x 2 3 x x x lim x 0 x 2 lim 3x 2 3x x x x 0 y y = x3 x 3x 2 The details here are a lot like those for the earlier example involving y = x2, so they should be fairly easy to follow. The initial goal is to simplify the numerator to be able to remove a factor of x from it. In the second last expression, both terms that contain x as a factor become negligible in the limit that x 0, leaving the single term result, 3x2. You can see from the small sketch of the graph of y = x3 just above to the right that this formula corresponds to the major features of the shape of the graph. The formula says that the slope of the graph is always a positive number, which is true, since that graph is always varying upwards to the right. It also says, that the Page 4 of 8 The Derivative David W. Sabo (1999) slope of the graph will be zero at x = 0, and that also is clear in the sketch. We also expect that the graph will get steeper the farther x is from zero, and that also shows up in the graph. Example 2: Use the -method to determine the derivative of y x with respect to x. Solution We have immediately that x x x d d y x lim x 0 dx dx x and we seem to be stymied, because there is no obvious way to factor a x out of the numerator to cancel the x in the denominator. The difficulty is that the only reference to x is buried inside one of the square roots in this difference of two square roots. However, you may recall a technique from basic algebra that can be used to turn a difference of two square roots into an expression with no square roots (used when it is necessary to "rationalize" the denominator of a quotient). In this case, we can use the same approach, but we will "rationalize" the numerator instead. Thus x x x d x lim x 0 dx x x x x x x x x x x x lim x 0 lim x 0 lim x 0 lim x 0 x x x x x x x x x x x x 1 x x x y 1 2 x y = x Thus, d 1 x dx 2 x x This formula should seem plausible in view of the shape of the graph of y x , shown above to the right. The graph appears to go vertical right at x = 0, consistent with the infinite value of the slope at x = 0 given by the derivative. Then, as x increases from zero, the square root of x in the denominator of the derivative will make the value of the derivative decrease gradually, and we see the original graph tending to rise less steeply the farther to the right you go. The value of derivative here is always positive, and the slope of the original graph is also always positive. Note that the fact that David W. Sabo (1999) The Derivative y x exists only for non-negative Page 5 of 8 values of x corresponds to the derivative existing only for non-negative values of x as well (though in the case of the derivative, even the value x = 0 is excluded). Example 3: Use the -method to determine the derivative of y 1 x with respect to x. Solution We have immediately that 1 1 d d 1 x x x y lim dx dx x x 0 x 1 x 1 x x x x x x x x lim x 0 x lim x 0 lim x 0 x x x x x x y x y = 1/x x x x x x x 1 1 2 x 0 x x x x lim Here, the -method leads immediately to a complex fraction. The first few steps above are required to reduce the original complex fraction to a simple fraction in simplest form. By the time that is accomplished, the sought-after factor of x in the numerator is obvious. As expected, this derivative formula gives an undefined result when x = 0. Notice that the graph of the original function is decreasing wherever it exists, and indeed, the derivative has only negative values. For values of x very close to zero, the slope of the graph is very steeply downwards, and the value of the derivative will be a numerically large negative number. For values of x well away from zero, the graph of the original function is nearly horizontal, and the derivative gives a value very close to zero. Example 4: Use the -method to determine the derivative of y = 1 with respect to x. Solution This is quite a simple function. Since the formula for y does not contain x at all, the value of y is the not affected by the value of x. Thus, the -method gives d 1 1 0 1 lim lim lim 0 0 x 0 x x 0 x x 0 dx We know that the graph of y = 1 is just a horizontal line at y = 1, so it isn't surprising that the formula for the slope of that graph is just the value 0. Page 6 of 8 The Derivative David W. Sabo (1999) A Suggestive Pattern: #1 These last five examples presented so far in this document have all involved functions which are simple powers of x. You can see a pattern in the results when they are written in a common power-of-x format: d x3 3x 2 dx d x 2 2 x1 dx d 12 1 12 x x 2 dx d x0 0x0 0 dx d x 1 1 x 2 dx You can see that in each case, the derivative is also a power of x, but with the exponent decreased by one from the original function. Thus, the derivative of x3 contains an x2, the derivative of x2 contains an x1 or x, the derivative of x1/2 contains x-1/2. Furthermore, the power of x in the derivative is multiplied by a numerical factor which appears to be equal to the exponent in the original function. That is, the derivative of x 3 contains a factor 3, the derivative of x2 contains a factor 2, the derivative of x-1 contains a factor of -1, and so on. In fact, this apparent pattern is not a coincidence. With a bit of work, it is possible to demonstrate that in general, d x n n x n 1 dx (3) We will resist the temptation to attempt to prove this formula in part or completely, but it is valid for any realvalued exponent n. With this formula, we now never have to use the -method to find the derivative of any power of x with respect to x. Example 5: Find the derivative of the function yx 5 3 with respect to x. Solution Believe it or not, it is not that hard to determine this derivative using the full-blown -method as done in the previous examples. However, even that amount of work is not necessary now that formula (3) is available. The requested derivative here has the form of formula (3) with n = 5/3. Thus, d 53 5 53 1 5 23 x x x 3 dx 3 A second important pattern is suggested by the following example. David W. Sabo (1999) The Derivative Page 7 of 8 Example 6: Use the -method to determine the derivative of y = 3x2 + 7x -2 with respect to x. Solution Although this function is a bit more complicated than the ones in the previous examples, in that it has more than one term, we can easily apply the standard procedure. The numerator of the quotient in the -method formula is formed by subtracting the expression of the function above from the expression obtained when x is replaced by x + x: 3 x x 2 7 x x 2 3 x 2 7 x 2 d 2 3 x 7 x 2 lim x 0 dx x 3 x 2 6 x x 3 x 2 7 x 7x 2 3 x 2 7 x 2 lim x 0 x 6 x x x 7x 2 lim x x 0 lim y x 6 x x 7 x 0 x lim 6x x 7 6x 7 x 0 This formula seems to reflect some of the major features of the graph of the function, sketched above to the right. In particular, notice that the derivative will be zero when 6x 7 0 x 7 x 1.17 6 and we observe that the vertex of the parabola, the point where the tangent line would be horizontal is just to the left of x = -1. To the left of this point, the slope of the parabola is negative, and the value of 6x + 7 is negative for x < -7/6. To the right of this point, the slope of the parabola is positive, and the value of 6x + 7 is positive for x > -7/6. Thus, this simple formula forming the derivative seems to reflect the shape of the graph of the original function. Notice a couple of other things about this result. The '6x' term in the derivative appears to originate with the 3x2 term in the original function. We know that the derivative of x2 is 2x, and so it looks as if the derivative of 3x2 is just three times the derivative of x2 or 6x. The derivative of x is just 1, and it appears from this then that the derivative of 7x is just seven times that or 7. In fact, we know that the function y = 7x is a straight line with slope 7, so it makes sense that the derivative of 7x is 7. Also, it appears that to get the derivative of a sum and/or difference of two or more terms, you just form the corresponding sum and/or difference of the derivatives of the individual terms. All of these conjectures are true, and we'll state them formally and illustrate them more fully in the next document. Page 8 of 8 The Derivative David W. Sabo (1999)