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Lecture

# Notes Differentiation

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School
University of Maryland
Department
Mathematics
Course
MATH 140
Professor
all
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 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 h→0 if the limit exists. Ex. ( ) 2 f x =x A function can fail to be differentiable if: 2 Notes – Differentiation Aleah Pisarz 9/24/12 – 10/3/12 Ex. x⟦x∧x>0 f (x)={ 2 x ∧x≤0 Is f  differentiable at 0? Does '(0) exist? 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. Notes – Differentiation Aleah Pisarz 9/24/12 – 10/3/12 3 Derivative rules and trigonometry: Power rule: f x =x →f x =nx ) x−1 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 g x f x −f x g x) ( ) (g (x = 2 [(x) ] (bottom)(derivativetop)−(top)(derivativebottom) 2 (bottom) Chain rule: Let  h(x)=f g(x) , ) then  h x =f) (g x ))g (x) 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
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