ࡱ> ?A>'` Nbjbj .j%vvvvZAZAZA8A|B{[2B.E"EEEHHHZZZZZZZ$\h_fZH H|HHHZvvEE5[MMMHjvEEZMHZMMVbV@&WEB }k&ZAHV JZDK[0{[Vx{_I2{_&W{_&W$HHMHHHHHZZKHHH{[HHHH6<<vvvvvv Section 2.4: The Chain Rule Practice HW from Larson Textbook (not to hand in) p. 122 # 1-23 odd, 33-97 odd In this section, we want to differentiate different type of function compositions. The rules displayed are all special cases of the chain rule, which we discuss later in the section. General Power Rule  EMBED Equation.3  Example 1: Differentiate EMBED Equation.3 . Solution: % Example 2: Differentiate EMBED Equation.3 . Solution: % Example 3: Differentiate EMBED Equation.3 . Solution: Using the exponent law  EMBED Equation.3 , start by rewriting the function as  EMBED Equation.3 . Then using the general power rule, we calculate the derivative as follows.  EMBED Equation.3  % Example 4: Differentiate EMBED Equation.3 . Solution: % Derivative of an Exponential Function for an Arbitrary Base Base e:  EMBED Equation.3  Arbitrary Base:  EMBED Equation.3  Example 5: Differentiate EMBED Equation.3 . Solution: % Chain Rule For Exponential Functions Base e:  EMBED Equation.3  Arbitrary Base:  EMBED Equation.3  Example 6: Differentiate EMBED Equation.3 . Solution: % Example 7: Differentiate EMBED Equation.3 . Solution: % Example 8: Differentiate EMBED Equation.3 . Solution: Since this function is a product of two functions of t, we need to use the product rule. When taking the derivative of the second function, we apply the chain rule applied to the exponential function of base e. This gives  EMBED Equation.3  % Chain Rule Used to take the derivative of a composition of two functions. Chain Rule  EMBED Equation.3  Note: The general power rule and chain rule for the exponential function are special cases of the Chain Rule. Example 9: Differentiate  EMBED Equation.3  using the general power rule and the chain rule. Solution: % Chain Rule Applied to the Trigonometric Functions  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Example 10: Differentiate EMBED Equation.3 . Solution: %     (4=Oklm$ % 9 : M N O P S ^ k l  ƿrc_W_Hj"F h-CJUVaJjh0vUh0vjhh-5EHUj"F h-CJUVaJ h5jh5Uh)5CJaJ h0v5h)h/95CJaJ h<~h1X hdp.\hshdp. hbhshs5CJ aJ h|5CJ aJ h_Ehs5CJ aJ hr85CJ aJ h 5CJ aJ h=5CJ aJ Olm$ % 8 9 Q R S $a$gdgds$a$gd,:$a$gds`NNN   6 8 ^ ` b d j " $ & ( . @ B p r  ܯ~v~gZv~v~jH hShSEHUjK hSCJUVaJjhSUhS hS5jh-h-EHUjJ$F h-CJUVaJjh-Uh- h-5j/h0vh-EHUj%#F h-CJUVaJjhUh h5 h0v5h0vjh0vUjh0vh-EHU   h j ~ , .  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Solution: % Example 12: Differentiate EMBED Equation.3 . Solution: % Example 13: Differentiate EMBED Equation.3 . Solution: % Formulas for the Derivative of Logarithmic Functions Natural Logarithm:  EMBED Equation.3   EMBED Equation.3  Logarithm of an Arbitrary Base:  EMBED Equation.3  Example 14: Differentiate  EMBED Equation.3  Solution: % Chain Rule for the Logarithmic Functions Natural Logarithm:  EMBED Equation.3   EMBED Equation.3  Logarithm of an Arbitrary Base:  EMBED Equation.3  Example 15: Differentiate EMBED Equation.3 . Solution: % Example 16: Differentiate EMBED Equation.3 . Solution: % Example 17: Differentiate EMBED Equation.3 . Solution: Using the chain rule formula for the natural logarithm of an absolute value  EMBED Equation.3 , we calculate the derivative as follows:  EMBED Equation.3  % Example 18: Differentiate EMBED Equation.3 . Solution: On this one, it is easier to use some properties of logarithms to rewrite the function before differentiating. 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