ÿþ<html> <head> <title>Albert van der Sel : derivative function.</title> <meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1"> </head> <body bgcolor="#D8D8D8" link="blue" alink="blue" vlink="blue"> <font color="black"> <h1>Basic arithmetic/calculus.<br> In the series: Note 5.<br> <h1>Subject: How to differentiate and determine the derivative function.</h1> Date : 28 Februari, 2016<br> Version: 0.3<br> By: Albert van der Sel<br> Doc. Number: Note 5.<br> For who: for beginners.<br> Remark: Please refresh the page to see any updates.<br> Status: Ready.<br> <hr/> <font color="black"> <font face="arial" size=2 color="black"> <br> <h3>This note is especially for beginners.<br> <br> Maybe you need to pick up "some" basic "mathematics" rather <I>quickly</I>.<br> So really..., my emphasis is on "rather <I>quickly</I>".<br> <br> So, I am really not sure of it, but I hope that this note can be of use.<br> Ofcourse, I hope you like my "style" and try the note anyway.</h3> <br> Preceding notes:<br> <br> <a href="arithmetic4.htm">Note 1: Basic Arithmetic.</a><br> <a href="linear_equations3.htm">Note 2: Linear Equations.</a><br> <a href="polynomials5.htm">Note 3: Quadratic Equations and polynomials.</a><br> <a href="sinecosine3.htm">Note 4: The sine/cosine functions.</a><br> <br> This note: Note 5: How to differentiate and determine the derivative function.<br> <br> Each note in this series, is build "on top" of the preceding ones.<br> Please be sure that you are on a "level" at least equivalent to the contents up to, and including, note 4.<br> <br> <br> <font color="blue"> <h1>1. Introduction to the "derivative function".</h1> <font color="black"> <font color="blue"> <h3>1.1 taking the "Limit" of a function for a certain "x"</h3> <font color="black"> Before we go discussing "differentials" I need to touch on the subject of "taking the limit". This is really easy to understand.<br> <br> Professional mathematics can be very "formal" and can be quite hard to read, even if the subject is realtively easy.<br> <br> Ofcourse I fully understand that the professional literature is way way way better than my notes, but my goal simply is,<br> that you grasp concepts <B>quickly...</B><br> <br> Suppose you have the well-behaved function f(x)=2x+3.<br> Now, if you want to know the value of f(x), for say x=3, then you simply fill in "3" into "f(x)" and calculate "2x3 + 3 = 9"<br> Here, it is easily done, since "f(x)=3x+2" is a <B>smooth continuous</B> function (no gaps, no asymptotic behaviour).<br> <br> You may also say this: <I>if "x" approaches "3" very closely, then to what "y" will f(x) go to?</I> In the example above,<br> f(x) will ofcourse neatly approach 9, if "x" approaches "3".<br> <br> Sometimes mathematicians also write it as:<br> <br> lim<sub>x-->3</sub> 2x+3 = 9<br> <br> However, when you have a function which is "not neat", or not defined, for a certain x, the "lim" notation will help to correctly<br> decsribe f(x) for that "x".<br> <br> Suppose we have the function <B>"f(x)=1/x"</B>. This function behaves rather nicely, except when "x" approaches '0'.<br> When "x" gets very large (positive or negative), then "y" simply slowly approaches 0, but that's all fine.<br> When "x" approaches "0", we run into problems, since mathematically "1/0" is not defined.<br> It's mathematically "not nice" to say: "f(0)", since division by zero is not defined (actually, it runs to infinity).<br> <br> Figure 1. f(x)=1/x<br> <br> <img src="differential_2.jpg" align="centre"/> <br> <br> If x approaches '0' from the postive x-axis side, then "y" goes to + infinity.<br> If x approaches '0' from the negative x-axis side, then "y" goes to - infinity.<br> <br> But, in the "limit" notation, it looks way better:<br> <br> lim<sub>x &#8595; 0</sub> 1/x --> infinity<br> <br> lim<sub>x &#8593; 0</sub> 1/x --> -infinity<br> <br> So, we are not saying "x" equals '0', but we say instead that "x" approaches '0'.<br> <br> But, for a nice, continuous functions, the "lim" notation simply means <B>the value of f(x), for a certain x.</B><br> Nothing special here !<br> That is, say that for a nice, continuous function, that "x" approaches "a", then:<br> <br> lim<sub>x-->a</sub> f(x) = f(a)<br> <br> We will mainly use this "normal" behaviour, instead of approaching "gaps" or asymptotes etc..<br> <br> Please note that for any smooth continuous function f(x), it holds that:<br> <br> lim<sub>h-->0</sub> f(x + h) --> f(x)<br> <br> Since, <I>if h really is extremely small</I>, then "x+h" is practically the same as "x", and f(x + h) is practically the same as f(x).<br> <br> <font color="blue"> <h3>1.2 Introducing &#916;f(x)<B>/</B>&#916;x </h3> <font color="black"> We have already seen some <B>functions</B>, for which holds that each "x" is mapped to one "y".<br> Think for example of a linear equation, y=ax + b, where that condition is certainly true.<br> <br> But it's true too, for a quadratic equation like y=ax<sup>2</sup> + bx + c, or, for polynomials in general.<br> <br> We always have silently assumed (so to speak), that "functions" are rather "smooth" too, meaning that there<br> are no "gaps", and there (usually) is no "asymptotic behaviour" in the sense that the function very quickly "runs"<br> to infinity. For an example of the latter one: you might take a look at the tangent function (tan(x)), discussed in note 4,<br> which shows such asymptotic behaviour when x gets near &pi;/2.<br> <br> A function which does not have such irregularities, like gaps, is often characterized as "a continuous function".<br> <br> When we have an equation like "y=x<sup>3</sup> - x", we also often say that y=<B>f</B>(x),<br> where the function "f(x)" then is the same as "x<sup>3</sup> - x".<br> <br> It's just important, especially in this note, to get used to the notation y=f(x), where f(x) can be any<br> sort of function.<br> <br> <font color="blue"> Question: suppose we have the equation y=ax+b, then what would be f(x)?<br> Answer: f(x)=ax+b<br> <font color="black"> <br> <B>What could be called a "core" idea of taking the differential of a function?</B><br> <br> The text <I>taking the differential</I> already says a bit what we are looking for.<br> <br> We essentially want to find <I>the rate of change of "y", which is the same as "f(x)"</I>, compared to the<br> <I>the rate of change of "x"</I>.<br> <br> Or: we want to find the "ratio" of the change of "f(x)", to the change of "x".<br> <br> Does this give us extra information? Yes, it does. Take a look at figure 2 below.<br> <br> Figure 2.<br> <br> <img src="differential_1.jpg" align="centre"/> <br> <br> <B>Case 1.</B> We see a blue line, "f(x)=3", which is constant. No matter at which "x" you are, "y" will always be "3".<br> Does this function posess any sort of "rate of change"? No. f(x) never changes so the rate of change=0.<br> <br> We might express the change of f(x) as &#916;f(x), and the change of x as &#916;x. Indeed, "&#916;" is a<br> universal symbol for "delta", meaning "change".<br> <br> In the case of the constant line f(x)=3, the ratio would be &#916;f(x)<B>/</B>&#916;x, and that is "0", since<br> &#916;f(x) is "0". The line is constant, so there is no change at all.<br> <br> <B>Case 2.</B> We also see the red line "f(x)=4x". So, if you change "x" by one, the change of y will always be four times as large.<br> Really. For example, if you are at x=0, and take 5 steps to the right, then you are at x=5 on the x-axis.<br> However, y=f(5)=20. So, x changed by 5, and the value of y changed by 20.<br> <br> But you could also have considered a small change in "x". Suppose, on the x-axis, you are at x=1.<br> Next, you go to x=1.1 (so the change is only "0.1"). The corresponding change in f(x) would then be "0.4".<br> <br> In this case (of f(x)=4x), you might decide that &#916;f(x)<B>/</B>&#916;x =4.<br> No matter what change in "x" you would consider, then the corresponding change in "f(x)" is 4 times as large.<br> You might say: alright, but wasn't it already "evident" in the function itself: y=4x ? True.<br> <br> In general, the ratio of the changes might be expressed as:<br> <br> <TABLE border=0> <TR> <TD> <font size=4 color="brown">ratio of the rate of change</B>=</TD> <TD> <font size=4 color="brown">&#916;f(x)<br> ------<br> &#916;x<br> </TD> <TD> <font size=4 color="brown">&nbsp; &nbsp; &nbsp; (equation 1)</TD> </TR> </TABLE> <br> I would like to re-write that a bit.<br> <br> A: If we would change "x" to "x+h", where "h" can be any value, then the change in x would be "h". That's evident.<br> <br> B: For the corresponding change in f(x), we can say that it has to be "f(x+h)" minus "f(x)".<br> <br> For the statement(B), we may not say that de difference in the function is "f(h)". Why not?<br> Well, above we have only considered simple lines. But suppose the function is a parabola.<br> In such a case, depending on where you are on the x-axis, the value of f(h) varies enormously.<br> We always need to consider the change of x, with a truly corresponding change of f(x).<br> <br> It means: you can always "pick" any "x" to start with, say a certain "x" denoted by "x<sub>1</sub>",<br> and then change "x<sub>1</sub>" to "x<sub>1</sub>+h".<br> But then we always have to consider the change in the values of "f(x<sub>1</sub>+h)" and "f(x<sub>1</sub>)",<br> thus with respect to that particular "x<sub>1</sub>".<br> <br> In general, the ratio of the changes might thus be expressed as:<br> <br> <TABLE border=0> <TR> <TD> <font size=4 color="brown">ratio of the rate of change</B>=</TD> <TD> <font size=4 color="brown">f(x + h) - f(x)<br> ----------------<br> &nbsp; &nbsp; &nbsp; &nbsp; h<br> </TD> <TD> <font size=4 color="brown">&nbsp; &nbsp; &nbsp; (equation 2)</TD> </TR> </TABLE> <br> In considering the ratio of changes as we have seen in the examples above, does it add to our knowledge?<br> <br> With the actual functions (the lines) that <I>we have seen sofar</I> (y=3 and y=4x), the addition in knowledge is not really great.<br> Ofcourse, when the ratio is "0", you can say that we thus deal with a line with a constant value.<br> And, when the ratio is "4" all the time (for every x), we can say that we thus deal with a line that always<br> "changes" 4 times as fast as "x".<br> <br> But it gets more impressive if we consider more complicated function. Let's study a a good example in chapter 3.<br> <br> <font color="blue"> <h3>1.3 The differential of a function, and the <I>"derivative" function</I>, of a function.</h3> <font color="black"> I hope you can see the following reasoning, with the aid of figure 3.<br> <br> <I>If</I> the difference between x and x+h is small, and thus also the difference between f(x) and f(x+h) is small too,<br> we can draw a straight line between those two points on the curve of f(x).<br> Note that this line is almost a "tangent-line', for that small neighborhood.<br> <br> In the example shown in figure 3, I arbitrarily choose for the function f(x)=x<sup>2</sup>, but it could have been<br> <I>any</I> continuous function.<br> <br> Figure 3. Tangent line, if "h" gets small.<br> <br> <img src="differential_3.jpg" align="centre"/> <br> <br> If h is really get very small, the line is going te become the <B>"tangent line"</B>, with a <B>"gradient"</B> (or slope),<br> which is very much the same as the gradient of f(x) for that local neighborhood.<br> <br> So, if "h" getting very, very small, we more and more end up with a true tangent line.<br> <br> So, let's try to calculate the "differential" (as was shown above), when h -> 0:<br> <br> <TABLE border=0> <TR> <TD> <font size=4 color="brown">lim <sub>h-->0</sub> </TD> <TD> <font size=4 color="brown">f(x + h) - f(x)<br> ----------------<br> &nbsp; &nbsp; &nbsp; &nbsp; h<br> </TD> </TR> </TABLE> <br> => <br> <TABLE border=0> <TR> <TD> <font size=4 color="brown">lim <sub>h-->0</sub> </TD> <TD> <font size=4 color="brown">(x + h)<sup>2</sup> - x<sup>2</sup><br> ----------------<br> &nbsp; &nbsp; &nbsp; &nbsp; h<br> </TD> </TR> </TABLE> <br> => <br> <TABLE border=0> <TR> <TD> <font size=4 color="brown">lim <sub>h-->0</sub> </TD> <TD> <font size=4 color="brown">x<sup>2</sup> +2xh -h<sup>2</sup> - x<sup>2</sup><br> ----------------<br> &nbsp; &nbsp; &nbsp; &nbsp; h<br> </TD> </TR> </TABLE> <br> the x<sup>2</sup> and - x<sup>2</sup>, will cancel each out, so we have:<br> <br> <br> <TABLE border=0> <TR> <TD> <font size=4 color="brown">lim <sub>h-->0</sub> </TD> <TD> <font size=4 color="brown">2xh -h<sup>2</sup><br> --------<br> &nbsp; &nbsp; h<br> </TD> </TR> </TABLE> <br> <font color="brown"> <h3>= 2x</h3> <font color="black"> Since 2xh/h - h<sup>2</sup>/h = 2x -h<br> <br> And, because h approaches '0', we end up with 2x.<br> <br> <B>Mind you, we have a great result here.</B> We did not made any assumptions on "x" itself, so<br> the derivation is valid for the whole of the "x-axis", thus for complete f(x).<br> <br> What we found is that for the function f(x)=x<sup>2</sup>, the gradient of tangent line at any "x",<br> is "2x".<br> <br> -So, if you want to know the gradient of the tangent line for, for example x=3, then that would be "6".<br> Thus, the tangent line itself would be parallel g(x)=6x.<br> <br> -And, if you want to know the gradient of the tangent line for, for example x=5, then that would be "10".<br> Thus, the tangent line itself would be parallel to g(x)=10x.<br> <br> -And, if you want to know the gradient of the tangent line for, for example x=8, then that would be "16".<br> Thus, the tangent line itself would be parallel to g(x)=16x.<br> <br> Indeed, the slope is getting steeper if "x" increases, as expected with this parabola. <br> In chapter 3 we will explore tangent lines further in detail.<br> <br> At this moment, it's important to understand that the <B>derivative function</B> of f(x)=x<sup>2</sup>,<br> turned out to be g(x)=2x. This itself is just an ordinary function.<br> <br> Here I only use "f" and "g" to be able to explicitly distinguish both functions.<br> But there already exists a way to denote both functions in a proper manner.<br> Most mathematicians have agreed to use this.<br> <font color="brown"> <h3>If f(x) is the function, then the derivative function is notated by f <B>'</B>(x)</h3> <font color="black"> Please note the <B>'</B> symbol, to denote the derivative function.<br> <br> In physics, and some other sciences, the "d/dx" (or "&#8706; / &#8706; x") is also often used:<br> <br> <TABLE border=0> <TR> <TD> <font size=4 color="brown">f '(x)=</TD> <TD> <font size=4 color="brown"> df(x)<br> ----<br> dx<br> </TD> <TD> <font size=4 color="brown"> &nbsp; &nbsp; &nbsp; (equation 3) </TD> </TR> </TABLE> <br> Actually, often the "&#8706;" symbol is used for functions having more than one variable, like f(x,y,x).<br> For functions depending on just one variable, like f(x), simply the letter "d" is used, which then leads to the d/dx notation.<br> <br> Note that equation 3, is actually the "infinitesemal" variant of equation 1, where &#916;x goes to "dx".<br> <br> Then read it as follows: we want to see <I>the change</I> of f(x) (the delta), compared to (as a ratio to)<br> <I>the corresponding change</I> of "x" (also a delta), whereas the delta is assumed to approach zero.<br> <br> <font color="black"> As said before, we will often use the "f '(x)" notation, to denote the derivative function.<br> <br> <font color="blue"> <h1>2. Methods for finding the derivative function.</h1> <font color="black"> Above we found that f '(x)=2x, is the <B>derivative function</B> for the parabola f(x)=x<sup>2</sup>.<br> <br> For many types of functions (like e.g. x<sup>3</sup> and higer degree, sin(x), etc..) it can be proven <I>how</I> to obtain<br> the derivative function. We have seen one example on how to do that, and really, all others go in a similar way.<br> So, we are not going <I>to prove</I> the method on how to obtain the derivative function for all those type.<br> And, it's really not neccessary.<br> <br> <font color="brown"> <h3>1. The derivative of a Linear equation:</h3> <font color="black"> Here, we know that f(x)=ax+b<br> <br> (1): Let's start with the simplest case: f(x)=c, or, what is the same, y=c, where "c" is some constant number.<br> So this is a "constant line" running parallel to the x-axis. It has no gradient (or slope),<br> and it does not change at all if "x" changes. See figure 1 for an example of y=c.<br> Since it has no gradient, we have:<br> <br> f(x)=c<br> <br> then<br> <br> f '(x)=0<br> <br> (2); In case of general linear function, we can say that it has a certain slope, ot gradient. This gradient is constant,<br> since the function is a line. Per definition, a line has a constant slope, isn't it?<br> So, here is how to obtain the derivative function:<br> <br> If:<br> <h3>f(x)=ax+b<br> <br> then<br> <br> f '(x)=a</h3> or in the d/dx notation:<br> <br> <TABLE border=0> <TR> <TD> <font size=4 color="brown">d ax+b<br> --------<br> dx<br> </TD> <TD>= a</TD> </TR> </TABLE> <br> Yes, indeed! The coefficient "a" determines the "angle" of that line with the x-axis, or in other words: it's gradient.<br> <br> In a way, we may say that a line is it's "own tangent line".<br> <br> <B><U>Example:</U></B><br> <br> f(x)=3x+2<br> f '(x)=3<br> <br> This means that the line 3x+2 has a gradient of "3", meaning that for each single step of "x", then "y" climbs 3 steps up.<br> <br> <B><U>Example:</U></B><br> <br> f(x)= -4x-6<br> f '(x)= -4<br> <br> Note the "-" signs. This means that the line -4x-6 has a gradient of -4, meaning that for each single step of "x",<br> then "y" sinks 4 steps down.<br> <br> <font color="brown"> <h3>2. The derivative of a polynomial of any degree:</h3> <font color="black"> Suppose we have the function:<br> <br> f(x) = ax<sup>n</sup> (where a is some constant number).<br> <br> The power "n" can be any integer, like n=3, or n=4 etc... Suppose we have n=3, then the function would be f(x) = ax<sup>3</sup> <br> <br> Then, using the method demonstrated in section 1.3, it can be proven that the derivative function is:<br> <br> <h3> If f(x) = ax<sup>n</sup><br> <br> then:<br> <br> f '(x) = an x<sup>n-1</sup></h3> or in the d/dx notation:<br> <br> <TABLE border=0> <TR> <TD> <font size=4 color="brown">d ax<sup>n</sup><br> ------<br> dx<br> </TD> <TD>= an x<sup>n-1</sup></TD> </TR> </TABLE> <br> <B><U>Example:</U></B><br> <br> f(x)= 4 x<sup>3</sup><br> <br> then<br> <br> f '(x)= 12 x<sup>2</sup><br> <br> <B><U>Example:</U></B><br> <br> f(x)= x<sup>2</sup><br> <br> then<br> <br> f '(x)= 2 x<br> <br> yes, this latter example we have derived ourselves in section 1.3.<br> <br> <font color="brown"> <h3>3. The derivative of a "sum" of functions:</h3> <font color="black"> What we mean is this: suppose we have f(x) + g(x).<br> Or if you like, suppose we have the function v(x) for which holds: v(x) = f(x) + g(x).<br> <br> Then how do we determine derivative function of v(x)?<br> <br> That's really simple: it's like this:<br> <br> If:<br> <br> v(x) = f(x) + g(x)<br> <br> then<br> <br> v '(x) = f '(x) + g '(x)<br> <br> So, simply find the individual derivative function, of each part of the sum.<br> <br> <B><U>Example:</U></B><br> <br> f(x) = 3 x<sup>4</sup> + 2 x<sup>2</sup><br> <br> then<br> <br> f(x) = 12 x<sup>3</sup> + 4 x<br> <br> <B><U>Example:</U></B><br> <br> f(x) = -2 x<sup>2</sup> + 2x<br> <br> then<br> <br> f '(x) = -4 x + 2<br> <br> <font color="brown"> <h3>4. The derivative of a "product" of functions:</h3> <font color="black"> Quite similar to (3), but this time we can write that v(x) = f(x) . g(x)<br> <br> Then how do we determine derivative function of v(x)?<br> <br> If:<br> <br> v(x) = f(x) g(x)<br> <br> then<br> <br> v '(x)= f '(x) g(x) + f(x) g '(x)<br> <br> <br> <B><U>Example:</U></B><br> <br> f(x) = 2x<sup>2</sup> . 2 x<sup>3</sup><br> <br> then<br> <br> f '(x) = 4x . 2 x<sup>3</sup> + 2x<sup>2</sup> . 6x<sup>2</sup> = 8 x<sup>4</sup> + 12 x<sup>4</sup> = 20 x<sup>4</sup><br> <br> <font color="brown"> <h3>5. The "chain" rule:</h3> <font color="black"> Suppose we have a function that can be viewed as:<br> <br> f(x)=u(v(x))<br> <br> So, we first have "v" operating on "x", then followed by "u" operating on "v(x)".<br> <br> This is not uncommon. Just think of for example f(x)=(x<sup>2</sup>-3)<sup>3</sup><br> <br> So, we can interpret it as: u=v<sup>3</sup>, while v=x<sup>2</sup>-3.<br> <br> It has been proven that:<br> <br> If f(x)= u(v(x)) then<br> f '(x) = u '(v(x)) . v '(x)<br> <br> <B><U>Example:</U></B><br> <br> Suppose we have:<br> <br> f(x)=(2x-3)<sup>5</sup><br> <br> If we treat it like this:<br> <br> v=(2x-3)<br> u=v<sup>5</sup><br> <br> Then using the upper rule, we find:<br> <br> f '(x) = 5(2x - 3)<sup>4</sup> . 2 = 10(2x - 3)<sup>4</sup><br> <br> <font color="brown"> <h3>6. The derivatives of sin(x) and cos(x):</h3> <font color="black"> The sin(x) and cos(x) functions are very important in math and science in general.<br> <br> Using the method demonstrated in section 1.3, it can be shown that:<br> <br> <h3>If f(x)=sin(x) then f '(x)=cos(x)<br> <br> If f(x)=cos(x) yhen f '(x)= -sin(x)</h3> <br> <font color="brown"> <h3>7. The derivatives of sin<sup>n</sup>(x) and cos<sup>n</sup>(x):</h3> <font color="black"> Thanks to subsection 6, we know what the derivatives of sin(x) and cos(x) are.<br> <br> But what are the derivatives of sin<sup>n</sup>(x) and cos<sup>n</sup>(x), where "n" is some power.<br> For example, if n=2, we would have sin<sup>2</sup>(x) and cos<sup>2</sup>(x).<br> <br> In all this sort of tasks of finding the derivatives, the chain rule must be used.<br> <br> Suppose we want to find the derivative of <B>y = cos<sup>2</sup>(x)</B>.<br> <br> Let u = cos x, so that y = u<sup>2</sup><br> <br> Thus y = (cos(x))<sup>2</sup> = f(g(x)).<br> <br> According to the chainrule:<br> <br> [f(g(x))]' = f'(g(x))g'(x)<br> <br> thus, if we exactly follow the chain rule:<br> <br> [f(g(x))]' = "2cos(x)sin(x).<br> <br> <font color="brown"> <h3>8. The derivatives of sin(x<sup>n</sup>):</h3> <font color="black"> For the cos variant, the argument goes the same way as shown below.<br> <br> Let's consider the situation where we need to find the derivative of sin(x<sup>2</sup>).<br> For higher powers, the method is exactly similar to the method below.<br> <br> We need to use the "chain rule" of subsection 5.<br> <br> Let f(u) = sin(u) and g(x) = x<sup>2</sup>.<br> <br> Thus y = sin(x<sup>2</sup>) = f(g(x)).<br> <br> According to the chainrule:<br> <br> [f(g(x))]' = f'(g(x))g'(x)<br> <br> thus, if we exactly follow the chain rule:<br> <br> [f(g(x))]' = cos(x<sup>2</sup>)(2x) = 2xcos(x<sup>2</sup>).<br> <br> <font color="blue"> <h1>3. The second derivative.</h1> <font color="black"> If we have f(x), then usually (except at gaps, asymptotes etc..), we can determine f '(x), or the <I>derivative</I> function.<br> <br> However, in general, we can also determine the <I>derivative</I> function of that <I>derivative</I> function.<br> <br> I mean, you might also say that f '(x) is the first <I>derivative</I> function.<br> But if f '(x) itself can be differentiated, then we may obtain the second <I>derivative</I> function f "(x) of f(x).<br> <br> Example:<br> <br> Suppose f(x)= 2 x<sup>3</sup> + 3x.<br> <br> Then:<br> <br> f '(x) = 6x<sup>2</sup> + 3<br> <br> And<br> <br> f "(x) = 12x<br> <br> We know that the first derivative is interpreted as the "gradient" (or slope) of the tangent line at f(x).<br> <br> The second derivative, may be interpreted as the "gradient" (or slope) of the tangent line at f <B>'</B>(x).<br> <br> Or, if we want to see that in the "d/dx" notation:<br> <br> <TABLE border=0> <TR> <TD> <font size=4 color="brown">f "(x)=</TD> <TD> <font size=4 color="brown">d<sup>2</sup> f(x)<br> ------<br> dx<sup>2</sup><br> </TD> </TR> </TABLE> <br> <br> What we have seen in this note is not the whole story, but for this note, it's quite enough.<br> I want my notes to be "fast", but not overwhelming....<br> It's way better to let the material of this note "sink in", and try some examples by yourself.<br> <br> <font color="blue"> <h1>4. How to analyze, or "investigate", a function.</h1> <font color="black"> In note 6, I will collect all theory needed to (what mathematicians call) <I>analyze a function</I>, by using<br> a good illustrative example.<br> <br> Here I mean, for example, how to find the intersection(s) with the x-axis, the intersection with the y-axis,<br> and "special points", like the "minima" and "maxima" of that function.<br> <br> For about those special points: we know that <I>if the gradient is '0'</I>, then the tangent line is parallel<br> to the x-axis, and it must be on a "hill" (maximum), or "crest" (minimum). Only at such point, the gradient (or slope) is then '0'.<br> <br> <B>So: How to analyze a function? Please see note 6.</B><br> <br> <br> <font face="arial" size=2 color="blue"> <h3>That's it ! Hope you liked it.<br> <br> The next note is a super quick intro in how to "analyze" a function.</h3> <font face="arial" size=2 color="black"> <br> <br> <br> <br> <br> </body> </html>