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Chapter%2012%20-lect - Chapter12 Ch. n n n n n 12. 12....

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Calculating the Derivative Chapter 12

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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 0 ' lim h f x h f x f x h + - =

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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  = 23, then  dy/dx  = 0

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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 - - - = = = - = -

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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 If  f(x) = u(x)  ±

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Chapter%2012%20-lect - Chapter12 Ch. n n n n n 12. 12....

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