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The chain Rule 1a (derivative of outside) (derivative of inside)
exemple fut (4x+3) 9 964x+3)*(47= 36(4x+3)
Chip = (x2+3x+7) hei=64x2+3x+77  (2x+3) 25 ปี
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gux) = sin(x24*7 cos(x29xJ62x47 double chain fey)= (It cos 2x)
f'(=261+ corax Xsinax) a sin 2x need its own the hor fcx=201 tasar) (sinax ) 2 4citas
Don't forget about the age old question of What leads the students to drop out?
2 6 n2x) Impliat differentickien
differentiate both sides if you differentiate ay term, multiply it @ sale for day We also discuss several other topics like Who are the three main gods of hindu?
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excerpledy of x2 + y2 =25
2x+2y=0 differen both sides H = 2x + Don't forget about the age old question of Which of the following is an example of an etic description of teen pregnancy in america?
3 y =ô .
y = 2x solve 1 exemple 2 Sinx up ay ty=10x Don't forget about the age old question of How to determine the original mass or spring stiffness?
exby+Y'**]+ y = 10 If you want to learn more check out It is one of the biggest contributing factors to every organization, what is it?
Product Rule  99%  2 + x + y = 10
+dy = 10+ 2xy = Loss dy Cx2+1)= 10 + 2xycosx.
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Sinxx +y= asХ Cosx[
xy + y x2] + dy = 20
Product Rule da Cos X 2xy  dy xa+ dy = 20
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 dyx+dys 20208 x + 2xy. dy (X2+1)= 20cost axy = (20cosx + 2xy.
(x2+1) Trig Derivatives a sint cosx id tanx= secay a CSc x= CSC Xcotx
& Cosx
 Sinx
cotx  CSC2x
& secx= secx tanx
dx
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Inverse
Trig
Derivatives
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Log Derivatives e & in x = t ta in [f(x] = I. fcx) a = ex exemple f(x)= 1hCx2+10)
into caso I f(x) = in [(2x+372 7 2 In C2x+3)2  InC 32) 6
[63x206 = 2 In (2x+3)  6 ln (3x2)
= 2(2x + 2)6 (373)
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log
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importante example find the deriv. of e2  ex. (3x) =Bex
Experial Desistives.
het ex d. at ateina de tor e for f(x) atempten) at an affo). Ina.f(x) to do
exemple (3*) 13. In 3
food ex.x2  product rule et x 2 + x et
Texx + 2xello froetx.sinar (x.sinan). CX sin2x + sinax x) chain rule
= ecx. sinds). Closin2x + cosax.2.X)
Fleksin** (sinax + cos2x+27 Linear Approx. main equation = L(x) = f(a) + f (a) (xa) i exemple find the linear approx of f(x)=x at a=4
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use the linear approx to estimate 9402 plug in 244,02) = 2 + +64.02) 1 = 12.005
Absoluk min/max  Local min/max in
if if
f(0) = fc) 2
fex) for all x in fcx) for all x in
Domain = absolute max Domain  absolute min
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example how to find the extreme values que tout
f(x) = x2+2x2 [2, 2] ayson 10 find critical numbers from (a,b).
 take deriv. 2x+
2 8 set to ot solve ax=2 X 2 3 evaluate fex) at critical numbers
f(1) = 12 + 2(1)2= 3 (3 evaluate fra) + f(b)
f(2) = 22 +26272   2.
FCb) = 2ataca 2  6 14 conclude: biggest # = max, smallest =min
max= 6 or (2,6)
2 min=3 (13) local min/max: point where a functions increased decrease changes exemple t x2+2x2
P tare derivative & find critical rumber at en
2x+2=0 crit = 1  use number line test test a value on both sides of crit. #
  f'(2)= (negative value)
f(0) =  (negative value)  the function stays – on sooth sides , so no
local min/max СХrpu KX) = х2
f'(x) = ax crito #io in low change in inasil decr. &
 f(1)=
sono f(1) =+ ti so draw with the line
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Ilocal min at
local min at
local min
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Good conditions to know Ex Value Theocn (EVS).
if fet) is continues on Caib] then thace exists als minimax values if there is not a closed intertal. EXT Hors not apply
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Fermat's Theorem  it fex) has a local max of min at a t it fc exists
fcc7zo
+2 =6 dous n = o fona a Esc m4x4 a function may have a local minimax where it is not diffentiable
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Mein Value Theoren ( S cond Wons if, fois o cookies om C
b) differkuble on Calo) then there exists atleast I point cin [ab] fkas= f(b)flad
exemple fex = 1 + 3x  1 [2,97 note that 2 a +b=a a determine continuity Demain = (09
19] = 62,92 Take the derivative
fox) = 1 + 25x1 & it cx1713 f'w=f(x17213 f'( =
3473
• Determine Domain of f'(x) )
cannot differentiate at the 1 f is differentiable (oo, DUCI ) which fits in interval Casa
us fcb)  F 62 f(a) = +
be 1 set derivative = to value found in step 2
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1+ (13)
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Rolls Theoom (MVT plus I extra condition) a) continues Saiba b) differentiable faib) c) flad fcb)
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Concavity / hflection Points
f" x 70  increasing f"(x240  decreasing €" >o = concuje ce f ) 20 = Concave down exemple fcx) = x2 + 3x29x+4
f'c = 3+ CX9
f"(x) = 6x+6 2 find critical numbers  6x+6=0 (3 Test with number line a
Sture
TO
f"(2): AFC) = + FH so concore up = (1,0)
concave down= (0017 Inflection Point  change of
concavity
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The change of concavity in couve example is so to find the full point use
(1, IL11) (lis)
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Derivative Graphs the information you need for a graph of a derivative
Course & sketch)
f(x)
Intercepts Asymptotes
use I find
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open intervals of incel decr. 7 local extreme values I
use f'(x) find
use suf Ex
Open intervals concow up down inflection points
L'harital's Rule & indeterminate Forms The 7 indeterminine forms 8. , 2000, 0.00,1,0, so l'hopitals rule works directly on 8 and 2 example lim stanza o apply l'hopituds rule
Y01 to 3% [l'hopitals rull
take decitative of top & bottan if the function is reduced to inteleoninale forms 8 to
keep taking the derivative cas long as its & until a value is reached 50+ lin 3ten Tx 2\sec?(78).  31.c03 033)
tot tanax 3 sec 2(3x) 3 cos (7x) Plug in lim alcoscio = 21.1= 7
X>0+ 3.205470)
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Cuptum.zation doyeckie: momize or amize What to do?
come up with a fex Codepends on the prooem) @ Take the derivative of to 3 Find critical s. ex and the dimensions of a rectangle with an area of loouma whose perimeter is as small as possible
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P= x+14+Y  P=1x + y
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We are trying to minimize
the perinder so take the
derivative
of p=2x+2y
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first change P=2x+2y into a single variable problem. given by Alw 1000m2= xy werde proto change it into y so y 1000 then plug into P
P= 2x+2/1000) = ax+ 2000 take derivative 2x+ 2000x*+ P = 22000 x 2 = 2 2000 Superor B_2000) @ Find critical numbers
check critical 2x+200050 Woo x2=0
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bring back 1000=xy. X=11000 so solve equation.
1000  Cookp X=ALOL "Z11000
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