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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