m(x) = f(x) + g(x) m (x) = f (x) + g (x) (The Sum Rule) n(x) = f(x) g(x) n (x) = f (x) g (x) (The Difference Rule) thing CFAIHIHD fkthf.
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1 Chapter 3 Differentiation Rules 31 Derivatives of Polynomials and Exponential Functions Aka The Short Cuts! Yay! f(x) c f (x) 0 g(x) x g (x) 1 h(x) x n h (x) n x n1 (The Power Rule) k(x) c f(x) k (x) c f (x) m(x) f(x) + g(x) m (x) f (x) + g (x) (The Sum Rule) n(x) f(x) g(x) n (x) f (x) g (x) (The Difference Rule) p(x) e x p (x) e x : Ex: Prove! # % & # % (&)!" et g ( x ) c f ( x ) gutting eaten c I in fkthffi# his 0 thing CFAIHIHD c f ' Cx ) Note: Proof of Power Rule on page 175 uses the Binomial Theorem
2 Ex: Differentiate (w4?xafx yttxtt ( xgttrd ' ' + a) y & # sx b) f( r), + / 0 's P 3# or x3ax 35 ) s afe3e 3 3X now f G) e' ' + xeorfcrkertre ftpextexetorftrterteret c) 1 " 2 y a 4 2 '4fz) ' '
3 BewMath0#abin FM square Bear Tjp#2yrxx 34M 3 I Run y y3y, 1 survives
4 Escape from New York 1 9 *r x 2 7 y? Fa Fix in 213ft
5 # FEI ZIZZIZZ 34 n F t± I 84 Zxy 23
6 Definition of e It s clear from the study of exponential functions that the slope of the tangent to f(x) a x at x 0 is 1 for some value of a between 2 and 3 Mathematicians wanted to find that value The slope of the tangent line at x 0 is given by the first derivative at 0: 9 8:6 ;9(8) < ;<> < ;? f (0) lim lim lim So the definition of e is 7 5 ;
7 MK 32 The Product and Quotient Rules 9 9: ; :? Ex: Find m (x) if m(x) x 3 x 4 Is the derivative of m(x) equal to the product of the derivatives of its factors? mcx )x3x4 7 ) 7 6 x' * xd(4 3)12 5 # 7 6
8 Geometric Explanation of the Product Rule (via Gottfried Wilhelm Leibniz, one of the two fathers of calculus along with Sir Isaac Newton) Start with f(x)g(x), which can be visualized as the white rectangle above Let s imagine that we change x by an amount equal to : Now we have a new product, f(x 0 + :) g(x 0 + :), which is the larger rectangle We see from the picture that Whole Rectangle White Rectangle + Side Blue + Top Blue + Grey f(x 0 + :)g(x 0 + :) f(x 0 )g(x 0 ) + f(x 0 ) + g(x 0 ) ;+ ; So the change in the product is just the blue and grey rectangles: ; f(x 0 + :)g(x 0 + :) f(x 0 )g(x 0 ) f(x 0 ) + g(x 0 ) ;+ ; If we divide the whole thing by :, we get (LM) L(N >) O " " + M(N >) 9 + ; O " " Then, if we let : 0, 9 9: ; : : ; : R : + : ; (:) (Note: ; f(x 0 + :) ;(:) à 0 as : à 0)
9 2e x The Product Rule K KL M L N L M L K KL N L + N L K KL M L See page 187 for a similar explanation of what happens with the quotient of two functions: The Quotient Rule K KL M L N(L) N L K KL M L M L [N L ] Q K KL N L Ex: Differentiate Lowd high highd low low low a) f(x) (x 2 2x)(1 + e x ) f ' ( x ) ( 2x)F a+e )+ ( ' x xtzx ) ( HE ) (xtz )( Hey ' + (xtzxj ( He ) fex)(xz2x)( 0+0+(2 2) ( HED e ' 2e x2e e +2 +2*22 ex 2e 2
10 f ( ) # f ( ) [gc D "
11 ZD b) g(x) "2 ;["?:0 \ Gtexkx ' gkxk ' ( ( ltexjz xtzx )( Hey ' ( HE )( 2 2 ) ' TIE 2e ( x?zd(e ) x2e t2xe ftp#ikei2e*@2cxy+eexxzixd
12 sinxfsnx 33 Derivatives of Trigonometric Functions y wstz Egypt ) (sin *) cos * )* ) (cos *) sin * )* IT 1 II Ex: Using the above, find the derivative of y tan x abc " y1 Gnxkcosx ) Co5 dea "! (tan *)!" sec[! * (cot *)!" csc[ *! (csc *) csc * cot *! (sec *) sec * tan *!"!" # # Csiwxkasxj ' ) 6s2x+sN2 tws#seix
13 34 The Chain Rule The Chain Rule If g is differentiable at x and f is differentiable at g(x), then the composite function F >?@ defined by F(x) f(g(x)) is differentiable at x and F' is given by the product F'(x) f'(g(x)) g'(x) In Leibniz notation, if y f(u) and u g(x) are both differentiable functions, then!j!j!k!"!k!" Ex: Differentiate 1) f(x) (x+1) 2 * f ' G) 2 # ' + 2) g(x) 3 * [ g ' G) deriugtlusiofe ' (1+0) der # iv inside of 2 ( + D ' I ( 3 xy 's 's G 2 ( x + D 2 +2 EYE xi f2x) or js ay 3) y sin (x 2 ) cos ( xd ZX 2 2 cos
14 1 Using the Chain Rule to come up with the derivatives of exponential functions:!!" E" E ", but what is!!" F"? First: a x (e ln a ) x Then:!!" F"!!" (Eno< ) " (E no< ) "!!" (KLF)* (Eno< ) " KLF a x ln a Question: What happens when a e? a a In a e e ex he e
15 2 E 35 Implicit Differentiation Ex: If y x 2 + 4, find!j!" d, Zx + O II x Ex: If y x 2 4, find!j!" G) Th 2 GY x date 2 Ex: If y 2 x 2 4, find!j 44!" + 2 x th (Hint: Use the Chain Rule), ay, a ) 2 y 2 0 Zy Zx,
16 4) Ex: a) Find y if x 2 + y 2 25 ky+ G4,tD /1 10 Zy x Zx E b) Find the slope of the tangent line to the curve at x 3 2ty225 (3,4 ) m y2l6 m y±4 ( 3, c) Looking at the graph, does part b make sense? Ėasies m
17 Inverse Trig Functions If x sin y, then y sin 1 x for r [ < y < r [ (prounounced arcsine ) But what s the derivative? Use x sin y, and solve for!j!" : X ( x ) SINY 1 cosy so!j!"? dea j??;sto 2 j??; " 2 cosy + s n2y 1 ( sny ) Derivatives of Inverse Trig Functions:!!" sin1 x?!?; " 2!" csc1 x? " " 2 ;?!!" cos1 x?!?; " 2!" sec1 x? " " 2 ;?!?!!" tan1 x?:" 2!" cot1 x??:" 2
18 This is Bob f ( x ) Bob is a f ( x ) guy y I threw a ball to Bob fyx ) I threw a ball to a I threw a red ball to guy y aguy ' day I threw a green ball to a guy day I threw a red ball to d d
19 f4x)y' M K } hsno fcb ) f(a ) ba Slopes give vs change Xz X f ( xth ) f Specifically the average of th for some h p difference quotient fcxz ) I f(, ) rate change xd th ) m f( X+h,fk#, th x ) f( flx ) And then limits! Ling f(x,t#)f( ) Average fcxith ) f(, ) change rate of Instantaneous velocity
20 tin tc 'th#k ) use charts h 0 find 's! And then Limit di mo*h high Tort Laws 's lim 1 / h 0 % h( '# lim ( ) xt m h o h o Xt 5 or 2h+5h+h2 ' ( x + 5th lim dijn ) h O 't 5
21 36 Derivatives of Logarithmic Functions What is!!" #$% <'? Start with y #$% < ', then ( j x * *' (j * *' (') ( j ln ( *1 *' 1 *1 *' 1 ( j ln ( So, 4109a!j??!" < xyz {\ c < " c < If y ln x, then!j!"? " c 0? " In general, * *' (ln <) < < " u is " special aftnsise Ex: Find f (x) if f(x) ln(x 3 + 1) f '(x) } Ex : g ( ) In (46 3 2) find ( ) 4*6 1 f4fsnx) g ' gtxk
22 KB Cool Tricks: Ex: Find y if y ln "+ : ~";B [" Ä ;1" Yuck this one looks horrible, until you remember that: ln "+ : ~";B ln [" Ä ;1" ('~ + 3' 5)? ln [ (2'1 4') e g *I i* In (F) ha xts 45k, In A' Una Ex: Find!!" '[" (Hint: Try this y ' [" ln y ln ' [" S ( ln y 2x ln x) 'T y, ± [mu] lny Y ' Zxlnx # ' 1 I n' + 2 and ' Zlnx + Zx ( Zlnx +2 y ' ( Zlnxtz )y ( Zlnxt D ZX zx " ( Inx + 1 )
23 37 Rates of Change in the Natural and Social Sciences Whoo hooo! Word problems!!!!! Ex: (#8) If a ball is thrown vertically upward with a velocity of 80 ft/s, then its height after t seconds is s(t) 80t 16t 2 a) What is the maximum height reached by the ball? µ vertex x ten Ff ' V ( t ) 80 32T 0 :# Instant t } Taft set b) What is the velocity of the ball when it is 96 feet above the ground on its way up? On its way down? Notes: Marginal Cost C (x) The change in cost when you produce one more item Velocity Gradient Instantaneous rate of change of velocity with respect to r dv/dr
24 SA )ax2+bx+c (E) 2ax+b v (f) O : 2axtbO bza b Xb
25 38 Exponential Growth and Decay Radioactive Decay Fact: Radioactive substances decay at a rate proportional to the remaining mass Simply speaking, a big pile decays faster than a small pile So, for m mass and t time,!ñ %&, for some constant k In other words, m(t) m 0 e kt, where m 0 is the initial mass!ö Ex: (#8) Strontium90 has a halflife of 28 days a) A sample has a mass of 50 mg initially Find a formula for the mass remaining after t days b) Find the mass remaining after 40 days c) How long does it take the sample to decay to a mass of 2 mg? d) Look at the graph of the function Does it make sense?
26 39 Related Rates This section offers applications of Implicit Differentiation We ll use our understanding of one change to compute another change possibly one that s hard to measure directly Ex: Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 cm 3 /s How fast is the radius of the balloon increasing in general? How about when the diameter reaches 50 cm? Hint: Volume of a Sphere 1 ~ )*~ We know:!â!ö 100 cm3 /s We want to find:!!ö
27 Ex: Car A is traveling west at 50 miles per hour and car B is traveling north at 60 miles per hour Both are headed for the intersection of the two roads At what rate are the cars approaching one another when car A is 03 miles and car B is 04 miles from the intersection? Ex: A water tank has the shape of an inverted circular cone with base radius 2m and height 4 m If water is being pumped into the tank at a rate of 2 m 3 /minute, find the rate at which the water level is rising when the water is 3 m deep Hint: Volume of a cone? ~ )*[ h, and use similar triangles to eliminate r
28 310 Linear Approximations and Differentials Let f(x) be a differentiable function What is the equation of line tangent to f(x) at x a? 1) We want the point (a, f(a)) to lie on the tangent line 2) We want the slope of the line to equal f (a) y y 1 m (x x 1 ) y f(a) f (a) (x a) y f(a) + f (a) (x a) This is called the linearization of f at a Ex: Find the linearization of f(x) x 3/4 at x 16
29 Differentials If f(x) is differentiable, then f (x)!j!" By multiplying both sides by dx, we get dy f (x) dx We call dx and dy differentials But what do dx and dy represent? Since f (x) gives us the slope of the tangent line, dy and dx represent values related to the tangent line On the other hand, x and y represent values related to the original curve, y f(x) Ex: Let f(x) x 3 + x 2 2x + 1 Compare y and dx when x changes from 2 to 205 f(205) f(2) So y f(205) f(2)
30 But dy f (x) dx f (x) f (2) dx x so dy Note that dy is slightly easier to calculate than y Ex: a) The radius of a sphere was measured and found to be 21 cm with a possible error in measurement of at most 005 cm What is the maximum error in using this value of the radius to compute the volume of the sphere? If the radius of the sphere is r, then its volume is V 1 ~ )*~ If the error in the measured value of r is denoted by dr r, how can we calculate V? Let s approximate it with dv: dv b) What s the relative error in this example? Relative error ã â!â â This can be expressed either as a decimal or a percentage
31 311 Hyperbolic Functions In the same way that sin x, cos x and tan x are related to the unit circle, x 2 + y 2 1, sinh x 0\ ; 0 ç\ [ cosh x 0\ : 0 ç\ [ csch x sech x? abcé "? deaé " tanh x abcé " deaé " coth x deaé " abcé " sinh, cosh and tanh are related to the hyperbola, x 2 y 2 1 In this case, though, the argument isn t equal to the angle, but to twice the area of the shaded region Ex: Find!!" sinh E
m(x) = f(x) + g(x) m (x) = f (x) + g (x) (The Sum Rule) n(x) = f(x) g(x) n (x) = f (x) g (x) (The Difference Rule) thing CFAIHIHD fkthf.
. Chapter 3 Differentiation Rules 3.1 Derivatives of Polynomials and Exponential Functions Aka The Short Cuts! Yay! f(x) c f (x) 0 g(x) x g (x) 1 h(x) x n h (x) n x n-1 (The Power Rule) k(x) c f(x) k (x)
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Chapter 3 Differentiation Rules 31 Derivatives of Polynomials and Exponential Functions Aka The Short Cuts! Yay! f(x) c f (x) 0 g(x) x g (x) 1 h(x) x n h (x) n x n1 (The Power Rule) k(x) c f(x) k (x) c
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Chapter 3 Differentiation Rules 3.1 Derivatives of Polynomials and Exponential Functions Aka The Short Cuts! Yay! f(x) = c f (x) = 0 g(x) = x g (x) = 1 h(x) = x n h (x) = n x n-1 (The Power Rule) k(x)
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