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Chapter 3 88

# Chapter 3 88 - —_——_— 238 CHAPTER 3 DlFFERENTIATION...

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Unformatted text preview: —__——________—_________ 238 CHAPTER 3 DlFFERENTIATION RULES 3 Review CONCEPT CHECK ————————————- . . d 1. (a) The Power Rule: If n is any real number. then Em”) : mun—1. The derivative of a variable base raised to a constant power is the power times the base raised to the power minus one. (b) The Constant Multiple Rule: If c is a constant and f is a differentiable function. then Ed— [cf(x)] : c di f(m). a: w The derivative of a constant times a function is the constant times the derivative of the function. (c) The Sum Rule: If f and g are both differentiable, then Eli [f(a:) + 9(a)] : di f(a:) + d1 g(m). The derivative :1: 9: ac of a sum of functions is the sum of the derivatives. (d) The Difference Rule: If f and g are both differentiable. then 5— [f (m) — g(a:)] : Eld— f (:13) — di g(m). The :c a: as derivative of a difference of functions is the difference of the derivatives. (e) The Product Rule: If f and g are both differentiable. then 6%: [f(ac)g(m)] : f(a:) (1%: g(a:) + g(m) Ed— ﬁx). a: The derivative of a product of two functions is the ﬁrst function times the derivative of the second function plus the second function times the derivative of the ﬁrst function. we] gm % ms) — M) d1 gtw) d (f) The Quotient Rule: If f and g are both differentiable. then —— [— _—’—’———-—— d9: 9W) lg(vc)l2 The derivative of a quotient of functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator. all divided by the square of the denominator. (g) The Chain Rule: If f and g are both differentiable and F : f o g is the composite function deﬁned by F(a:) = f(g(:r:)). then F is differentiable and F' is given by the product F’(a:) : f’(g(m))g’(a:). The derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. 2. (a) y : 1:" :> y’ : narTLTl (b) y : e“ :> y’ : ex (C)y:a\$ => y'zaxlna (d)y:lnw :> 3/21/93 (e)y:logaar :> ylzl/(znlna) (f)y:sina: => y':cosa: (g) y : cosx :> y' = — sinzc (h) y = tanw => 1/: sec2 3: (i)y:csca; :> y’:—cscm cots: Q)y=secav => y'zsecmtanm (k) y : cota: :> y’ : -csc2 a: (l) y : sin'1 a: => 3/ 2 UN (m) y : cos_1 as :> y' : —1/\/1—-'? (n) y : tan—1 x ﬁ 3/ : 1/(1 + 1:2) (0) y : sinhzc :> y' : coshac (P) y : coshac :> y’ : sinhm (q) y : tanhm :> y' : sech2 a: (r) y : sinh’1 \$ :> y’ : Um (s)y:coshﬁ1x => y':1/\/a:'2—-.1 (t)y:tanh'1a: :> y’:1/(1—3:2) eh —1 3. (a) e is the number such that 113“?) h : 1. (b) e z timou + :01” (c) The differentiation formula for y = am [y’ : am in a] is simplest when a = 8 because in e : 1. (d) The differentiation formula for y : loga m [y' = 1/ (m In a)] is simplest when a : 6 because in e : 1. ...
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