# Chapter 2(A) - Chapter 2 Differentiation Outline n...

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Chapter 2 Differentiation

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Outline n Derivative q Definitions q Rules of Differentiation n Other Types of Differentiation q Parametric Differentiation q Implicit Differentiation q Higher Order Derivatives
n Maxima and Minima q Local and absolute extremes q Finding Extreme Values q Critical Points q Increasing and Decreasing Functions n Derivative Test q First Derivative Test for Local Extremes q Concavity and Points of Inflection q Second Derivative Test for Local Extremes n Optimization Problems q Absolute Extreme Values

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n Indeterminate Form (Limits) q L’Hospital’s Rule q Other Indeterminate Forms
Derivative

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n yx = 1 n dy nx dx - = () n f xx = 1 ' n f x nx - = The derivative of with respect to . The derivative of ( ) with respect to . f with respect to --- w.r.t Derivative
Some results Question: How to derive these results? Using limits

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3. Derivative --- instantaneous rate of change of the function 2. Derivative --- (Geometrically) Gradient (Slope) of the tangent line 1. Derivative --- using the concept of limit ( ) () ' ( ) lim xa f x fa - = -
x y c y m xc =+ ------ gradient (slope) of the line m ------ intercept cy -

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x y y m xc =+ 21 gradient (slope) yy m xx - == - 11 (,) Axy 22 Bxy Straight line --- find gradient 1 x 2 x
a x y () y fx = gradient at = gradient of tangent line at ( ) ' ( ) lim xa AA f x fa - == - ( , ( )) Aafa tangent line at A Curve --- to find gradient

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a x y () y fx = ( , ( )) Aafa Question: Why we define ( ) ' ( ) lim xa f x fa - = - tangent line at A
a x y () y fx = ( , ( )) Aafa ( )) Bxfx x 21 gradient of line ( ) = yy AB xx f x fa xa - = - - - gradient at gradient of line A AB gradient at = gradient of tangent line at AA tangent line at A Why we define ( ) ' ( ) lim f x - = -

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a x y () y fx = ( , ( )) Aafa ( )) Bxfx x 21 gradient of line ( ) = yy AB xx f x fa xa - = - - - If we choose B close to A, then gradient of line gradient at A BA ; Choosing closer and closer to is the same as letting approaches . Taking limit, we have, gradient at lim(gradient of ) ( ) lim A AB f x = - = - tangent line at A Why we define ( ) ' ( ) lim f x - = -
y x 4 34 yx =+ Pause and Think !!!

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Chapter 2(A) - Chapter 2 Differentiation Outline n...

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