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Notes Differentiation WITH THE ANSWERS

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Department
Mathematics
Course
MATH 140
Professor
All Professors
Semester
Fall

Description
Notes – Differentiation Aleah Pisarz 9/24/12 – 10/3/12 1 Basic definition of derivatives: The rate of change in the values of  f  as  x  moves from  a  to  a+h  is lim f(a+h −f a ) h→0 h To make this “average rate” instantaneous one takes the limit of the above ration as  h→0 . If this limit exist, it is going to measure the instantaneous rate of change of  f '  at  a . It is called the derivative of   at  a  and is denoted by  f '(a) . Thus ' f(a+h )−f (a) f a =lim h→0 h if the limit exists. Ex. 2 f x =x ' f (a+h)−f (a) (a+h)−(a) 2 f (a)=lim =lim =¿ h→0 h h→0 h lim a +2ah+h −a 2 lim h(2a+h) h→0 = h→0 =lim (2a+h)=2a h h h→0 A function can fail to be differentiable if: 2 Notes – Differentiation Aleah Pisarz 9/24/12 – 10/3/12 Ex. f (x)= x⟦x⟧∧x>0 { x ∧x≤0 Is  f  differentiable at 0? Does f '(0)  exist? ' f 0+h −f (0 ) f h −0 f(h) f 0 =lh→0 h =lh→0 h =lh→0 h *at this point, we need to consider 1­sided limits −¿ h→0 (−1)=−1 h→0 −¿⟦h⟧=lim ¿ ¿ −¿ h⟦h⟧ h→0 h =li¿ ¿ h→0 −¿ f (h=lim ¿ h ¿ lim ¿ ¿ h→0 h=0¿ 2 h→0 +¿h =lim ¿ h ¿ f (h) h→0 +¿ =lim ¿ h ¿ lim ¿ ¿ the limit DNE, so no, f is not differentiable Is f continuous at 0? −¿ h→0 x ⟦x⟧=0 h→0 f (x)=lim ¿ ¿ lim ¿ ¿ +¿ 2 h→+¿x =0 h→0 f (x)=lim ¿ ¿ lim ¿ ¿ f (0)=0 Notes – Differentiation Aleah Pisarz 9/24/12 – 10/3/12 3 Yes, f is continuous at 0 Ex. f (x)=∣x∣ f x = −x∧x<0 { x∧x≥0 f is continuous but not differentiable If a function is differentiable at a, then it is continuous at a. However, if a function is  continuous at a, it is not necessarily differentiable at a. Derivative rules and trigonometry: Power rule: n ' x−1 f x =x →f x =nx ) Constant multiple rule: n ' n−1 f x =ax →f x =anx) Product rule: fg x =f x g x +g x f '(x) (first)(derivativeof second)+(second)(derivativefirst) Quotient rule: ' ' f (x = g x f x −f x g x ( ) (g [g (x)] (bottom )(deriv ativetop )− (top)(derivativebottom ) 2 bottom ) Chain rule: Let  h(x)=f g(x) ,) then  h x =f 'g x ))g (x) 4 Notes – Differentiation Aleah Pisarz 9/24/12 – 10/3/12 Trigonometric derivatives: sinx→cosx   cscx→−cscxcotx   cosx→−sinx   secx→secxtanx   2 2 tanx→sec x   cot→−csc x   Inverse trigonometric derivatives: −1 1 −1 1 sin x∨arcsinx→ 2 tan x∨arctanx→ 2 √1−x 1+x Exponential and logarithmic u u ' x2 x2 e →e u e →e 2x 1 2 1 lnx→ xx' ln x → 2 2x x 1 1 logax→ x' log53x→ 3 xlna 3xln5 2 →2 u ln2 2 →2 3ln2
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