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Unformatted text preview: Chapter 3: Topics in Differentiation Summary: Having investigated the derivatives of common functions in Chap ter 2 (i.e., polynomials, rational functions, trigonometric functions, and their com binations), this chapter begins by asking if the slope of a general curve can be found even if the curve does not have an explicit function representation. Instead, the assumption here is that the dependent variable y is implicitly described by an equation involving both x and y but where y = f ( x ) is not known. This process of finding dy / dx then allows one to find tangent lines to a curve even if an explicit description of the curve such as y = f ( x ) is not known. Implicit differentiation then leads to an extended version of the power rule, deriva tives of exponential functions, and also derivatives of the inverse trigonometric functions. Along the way, derivatives of logarithms are developed based upon a limit given in Chapter 2 and the derivatives of inverse functions are discussed based upon the reflective properties of f and f − 1 across the line y = x . Finally, the chapter concludes with a discussion of L’Hˆopital’s rule which is useful for evaluating limits that have an appropriate indeterminate form. OBJECTIVES: After reading and working through this chapter you should be able to do the following: 1. Find the slope of a curve that is defined implicitly (§3 . 1). 2. Take derivatives of logarithmic functions (§3 . 2). 3. Take derivatives of exponential functions and inverse trigonometric func tions (§3 . 3). 4. Use logarithmic differentiation to find derivatives of complicated functions (§3 . 3). 5. Use derivatives to relate the rates of different quantities that depend upon the same parameter such as time (§3 . 4). 6. Approximate functions using linearizations or differentials (§3 . 5). 7. Use L’Hˆopital’s rule to evaluate limits that have indeterminate forms such as , ∞ ∞ , 0 · ∞ , ∞ − ∞ , 1 ∞ , 0 and ∞ (§3 . 6). 49 50 3.1 Implicit Differentiation PURPOSE: To give rules for expressing the derivative of a func tion that is written implicitly. This section approaches the problem of being able to find the slope of a curve even if an explicit formula for the curve is unknown. In other words, most curves are implicit means that y = f ( x ) is not known described by a formula of the form y = f ( x ) . However, it may not always be prac tical or possible to find such an expression for y (i.e., consider y + sin ( xy ) = 3). In such cases, the chain rule can be used to find the slope dy / dx . The basic idea in this section is to assume that y is a function of x , that is y = f ( x ) , Implicit Differentiation 1. take derivatives using the chain rule 2. solve for dy / dx and then to differentiate the relationship describing the curve with respect to x by using the chain rule. In the end, the derivative dy / dx may be expressed in terms of x and y ....
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 Spring '10
 lee
 Derivative, dx

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