ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019

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1 ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 6. Differential Calculus 6.. Differentiation from First Principles. In tis capter, we will introduce te concept of differentiation and indicate wat it does. Tis capter and te net net one on integral calculus are introductory capters and we will build on tem in te second semester, so it is important to be comfortable wit te material before ten. We will start wit te formal definition and ten eplain wat it means. Definition 6.. (Derivative). Let f: (a, b) R, ten te derivative of f at (a,b) is defined to be () f f(+) f() (), if tis limit eists. Wen we find te derivative of a function, we say we differentiate it. Te process is called differentiation. Remark In te definition tere are te words if tis limit eists. If te limit doesn t eist (we won t look at ow tis can appen ere) ten te derivative of f does not eist at. Tere are many functions tat are not differentiable but we won t study tese in tis course. Let us now eamine wat te definition means. If we look at Figure, we will see tat te slope of te line (See Capter 2) from te point (,f()) to te point (+,f(+)) is just f(+) f(). Wen we differentiate a function at a point, wat we are really doing is to make smaller and smaller in f(+) f() and see wat appens to it (tis is wat te lim is telling us to do). Wat tis means grapically is sown in Figure 2. Hopefully tis figure will convince you tat te derivative of f at is te slope of

2 Figure. Slope of a line connecting (,f()) to (+,f(+)). te tangent line to te function f at te point. Tis also tells us wen a function is not differentiable. At any point were te tangent line does not eist (for eample were te grap as a kink in it) te derivative doesn t eist eiter. Figure 2. Geometric meaning of te derivative. Dr. Jon Seekey as prepared an interactive GeoGebra workseet wic sows te process of finding a derivative. It can be found at ttp:// Remark Often te derivative of a function f will be denoted by dy, df, d (f) or even f rater tan f (). Also note tat sometimes completely different 2

3 letters may be used, so you may see tings like g () or dg or indeed dy, were te roles of and y ave been reversed. All tese different notations mean eactly te same ting. Tey only eist since calculus was developed by different matematicians and te various notations ave persisted. Wile all tis may seem like a lot of effort to go to just to find te gradient of a tangent to a curve, it is etremely important since it arises in numerous different areas. Wenever you want to find te rate of cange of someting ten calculus will come in andy. For eample, if you ave a function representing te position of an object, ten te derivative will represent te velocity of te object. Similarly if you ave a function representing te velocity of an object ten te derivative of tis function will represent te acceleration of te object. Now let us differentiate some functions from first principles. Tis means tat we are going to use () to perform te differentiation, rater tan a table of derivatives. Eample () Find te derivative of te function f() = 3. Using (), we ave f () f(+) f() 3(+) = 3. 3 Tus f () = 3. (2) Find te derivative of te function f() = 2. Using (), we ave f f(+) f() () (+) = 2. 3

4 Tus f () = 2. (3) Find te derivative of te function f() = 2 3. Using (), we ave f f(+) f() () 2(+) ( ) = 6 2. Tus f () = 6 2. (4) Find te derivative of te function f() = Using (), we ave f () f(+) f() 3(+) 2 2(+)+ (3 2 2+) 3( ) 2(+)+ (3 2 2+) = 6 2. Tus f () = 6 2. (5) Find te derivative of te function f() = c, were c is a constant. Using (), we ave f () f(+) f() c c 0 = 0. 4

5 Tus f () = Some Common Derivatives. From now on, we will concentrate on te actual mecanics of differentiation, rater tan worrying about differentiating functions from first principles. In Table tere is a list of derivatives tat you sould be able to use. Note tat a formula seet will be provided in te eam, so you sould concentrate on learning ow to use tem, not on memorising tem. In te table represents a variable wile a represents a constant. f() f () Comments c 0 Here c is any real number n e a ln(a) sin(a) cos(a) n n ae a Here we must ave a > 0 a cos(a) a sin(a) Note te cange of sign Table. Some common derivatives Warning () Note tat te derivative of ln(a) is, no matter wat te value of a is (provided a > 0). Tis is NOT a typo. (2) Also note tat te derivatives of sin(a) and cos(a) are only valid if is in radians. If is in degrees ten etra constants would be needed but in practice we NEVER use degrees wen differentiating. As usual, a few eamples will make tings clearer. Please see Table Te Sum and Multiple Rules. Altoug te list of derivatives in Table is very useful, we would not get very far if tese were te only functions we could differentiate. Luckily tere are rules tat allow us to differentiate more complicated functions. Te first of tese allows us to differentiate sums of functions. Teorem 6.3. (Te Sum Rule for Differentiation). Let f: (a,b) R and g: (a,b) R, ten te derivative of f +g at (a,b) is given by provided tese derivatives eist. (f +g) () = f ()+g (), All tis says is tat if we want to differentiate a sum of two functions ten all we ave to do is differentiate tem separately and add te derivatives. 5

6 f() f () Comments 0 0 Note te derivative of 0 is π 0 π is just a number e 0 e is just a number cos() 0 cos() is just a number Since =, n = giving 0 = Here we take n = = 4 5 Here we take n = 4 π π π π is just a number e e e = e e is just a number e+ e e Here we take a = e 5 5e 5 Here we take a = 5 e 7 7e 7 Here we take a = 7 e e e e e = e e+ Here we take a = e ln() Here we must ave > 0 ln(5) Here we must ave > 0 ln( 5) Here we must ave < 0 sin() cos() Here we take a = sin(3) 3cos(3) Here we take a = 3 sin( 2) 2 cos( 2) Here we take a = 2 sin( π) π cos( π) Here we take a = π cos() sin() Here we take a = cos(4) 4sin(4) Here we take a = 4 cos( 5) 5 sin( 5) Note ( 5) = +5 cos(π) πsin(π) Here we take a = π Table 2. Some eamples of derivatives Here are a couple of eamples of te use of te Sum Rule. Eample () Find te derivative of f() = 2 +sin(2). f () = d (2 )+ d (sin(2)) = 2+2cos(2). (2) Find te derivative of f() = ln(2)+e 3. Provided > 0 (so tat te derivative of te first term eists), f () = d (ln(2))+ d (e 3 ) = 3e 3. 6

7 Te second rule tat will enable us to differentiate a larger range of functions is te Multiple Rule. Teorem (Te Multiple Rule for Differentiation). Let f: (a, b) R and c R, ten te derivative of cf at (a,b) is given by provided te derivative of f eists. (cf) () = cf (), All tis says is tat if we want to differentiate a constant multiple of a function, ten all we ave to do is first differentiate te function and ten multiply by te constant. Here are a couple of eamples of te Multiple Rule. Eample () Find te derivative of f() = 5 3. f () = 5 d (3 ) = = 5 2. (2) Find te derivative of f() = 3cos(2). f () = 3 d (cos(2)) = 3 ( 2sin(2)) = 6sin(2). Warning Te Multiple Rule can only be used to differentiate a product of a number and a function. If we want to differentiate te product of two functions, ten we ave to use te Product Rule wic we will study in te second semester. Of course we are free to use bot te Sum and Multiple Rules to differentiate a function and te following are a couple of eamples of tis. Eample () Find te derivative of f() = f () = d (52 )+ d ( 4)+ d (3) (using te Sum Rule) = 5 d (2 ) 4 d ()+ d (3) (using te Multiple Rule) = 5(2) 4()+0 =

8 (2) Find te derivative of f() = e 2 2cos( 3). f () = d ( e 2 )+ d ( 2cos( 3)) (using te Sum Rule) = d (e 2 )+( 2) d (cos( 3)) (using te Multiple Rule) = ( 2e 2 ) 2( 3( sin( 3))) = 2e 2 6sin( 3). 8