Chapter%2012%20-lect - Calculating the Derivative Chapter...

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Unformatted text preview: Calculating the Derivative Chapter 12 Ch. 12 Calculating the Derivative n 12.1 Techniques for Finding Derivatives n 12.2 Derivatives of Products and Quotients n 12.3 The Chain Rule n 12.4 Derivatives of Exponential Functions n 12.5 Derivatives of Logarithmic Functions 12.1 Techniques for Finding the Derivative n Using the definition (below) to calculate the derivative can be very involved. n Section 11.5 introduces rules to simplify the calculation of derivatives. n The simplified process does not change the interpretation of the derivative . ( 29 ( 29 ( 29 ' lim h f x h f x f x h +- = Techniques for Finding the Derivative n Each of the above represents the derivative of the function f(x) (or y ) with respect to x. NOTATIONS FOR THE DERIVATIVE The derivative of y = f(x) may be written in any of the following ways: ( 29 ( 29 ( 29 ' , , , x dy d f x f x D f x dx dx CONSTANT RULE If f(x) = k , where k is any real number, then f (x) = 0 (The derivative of a constant is 0) If f(x) = 12, then f (x) = 0 If p(t) = p , then Dt [ p(t) ] = 0 If y = 23, then dy/dx = 0 POWER RULE If f(x) = xn for any real number n , then f (x) = nxn 1 (The derivative of f(x) = xn is found by multiplying by the exponent n and decreasing the exponent on x by 1.) If y = x 3, then Dxy = 3 x 3 1 = 3 x 2 If y = x , then dy/dx = 1 x 1 1 = x 0 = 1 1 2 1 2 1 1 2 1 1 If (rewritten ), then 2 2 dy y z y z z z dz-- = = = = 1 2 1 1 2 2 z z = = CONSTANT TIMES A FUNCTION Let k be a real number. If f (x) exists, then Dx[kf(x)] = kf (x) (The derivative of a constant times a function is the constant times the derivative of the function.) ( 29 ( 29 3 3 2 2 If 4 , then 4 4 3 12 dy dy y x x x x dx dx = = = = ( 29 3 2 3 2 1 2 1 2 3 If 10 , then 10 10 15 2 dy dy y x x x x dx dx = = = = ( 29 1 2 2 5 If , (rewritten 5 ), then 5 1 5 dq q q z z z z dz--- = = =- = - A FUNCTION DIVIDED BY A CONSTANT Let k be a real number. If f (x) exists, then (The derivative of a function divided by a constant is the derivative of the function divided by the constant.) ( 29 ( 29 ' x f x f x D k k = 2 5000 5000 2 if , then ' 100 100 x x x y y-- = = SUM OR DIFFERENCE RULE...
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This note was uploaded on 01/12/2012 for the course ECO 3401 taught by Professor Staff during the Fall '08 term at University of Central Florida.

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Chapter%2012%20-lect - Calculating the Derivative Chapter...

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