lect116_9rev_f11

lect116_9rev_f11 - Example If f ( x ) = ( x 2 + 1) find f (...

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Thursday, September 29 Lecture 9 : Differentiation: The Chain rule (Refers to section 3.6 in your text) Students who have mastered the content of this lecture know about : The Chain rule. Students who have practiced the techniques presented in this lecture will be able to : Apply the Chain rule correctly when differentiating a composition of functions. 9.1 Introduction To determine the derivative of a composition of functions f ( g ( x )) using the definition of the derivative requires a tedious procedure involving limit computations Fortunately there exists, for many composed functions, a method of obtaining the derivative without solving a limit expression. This method is called the Chain rule . 9.2 Theorem – The Chain rule . If g is differentiable at x and f is differentiable at u = g ( x ) then the composite function F = f ° g defined by F ( x ) = f ( u ) = f ( g ( x )) is differentiable at x and F is found by computing the product Equivalently, if y = f ( u ) and u = g ( x ) then Proof is omitted. 9.2.1
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Unformatted text preview: Example If f ( x ) = ( x 2 + 1) find f ( x ). 9.2.2 Example Differentiate the function ( x 3 1) 100 . 9.2.3 Example Find the derivative of 9.2.4 Example Find the derivative of the function 9.2.5 Exercise Determine dy / dx if 9.2.6 Example Suppose y = f 1 ( x ). Find an expression for ( f 1 ( x )) by using the Chain rule. 9.2.7 Example Suppose p ( t ) = t 3 2 represents the position of an object moving in a straight line at time t . The instantaneous velocity v ( t ) at time t is known to be the instantaneous rate of change of the position p ( t ) at t , that is, v ( t ) = p ( t ). The acceleration a ( t ) at time t is defined to be the instantaneous rate of change of the velocity v ( t ), that is, a ( t ) = v ( t ). Find the acceleration at time t = 3. So a (3) = 18 units. (No chain rule is involved in this question, just the process of differentiation applied twice successively.)...
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lect116_9rev_f11 - Example If f ( x ) = ( x 2 + 1) find f (...

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