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MATH 2413Fall2012, Lectures17-19 QingwenHu DepartmentofMathematicalSciences TheUniversityofTexasatDallas Richardson, Texas [email protected] October, 2012 . QingwenHu (UTD) Lectures17-19 . . . . October, 2012 . 1/4 OutlineofSections3.3–3.5 1. Speciallimits lim h→0 sin h cos h − 1 = 1, lim = 0. h→0 h h 2. Derivativesoftrigonometricfunctions: e.g., d d (sin x) = cos x, (cos x) = − sin x, dx dx d 1 (tan x) = = sec2 x dx cos2 x 3. TheChainrule: If y = f (u) and u = g(x) aredifferentiable, then d ′ ′ dx f (g(x)) = f (g(x))g (x). i.e., dy dy d u = dx d u dx 4. TheexponentialRule: from d x dx a = ax ln a, a > 0, whichcanbederived ax = eln a = ex ln a . x . QingwenHu (UTD) Lectures17-19 . . . . October, 2012 . 1/4 5. TheexponentialrulecombinedwiththeChainRule: If y = f (x) isdifferentiable, then d f (x) e = ef (x) f ′ (x). dx 6. ApplyingtheChainrulemultipletimes: e.g., If y = f (u), u = g(v) and v = h(x) aredifferentiable, then d f (g(h(x))) = f ′ (g(h(x)))g ′ (h(x))h′ (x). dx i.e., dy dy du dv = . dx d u dv d x 7. Implicitdifferentiation. . QingwenHu (UTD) Lectures17-19 . . . . October, 2012 . 2/4 8. Derivativesofinversetrigonometricfunctions: d 1 arcsin x = √ dx 1 − x2 d 1 arccos x = − √ dx 1 − x2 d 1 arctan x = dx 1 + x2 . QingwenHu (UTD) Lectures17-19 . . . . October, 2012 . 3/4 Exercises: 1. Define f : R → R by { f (x) = if x ≤ 1, ax + b if x > 1. x2 Findthevaluesof (a, b) sothat f isdifferentiableeverywhere. Isthe valueof (a, b) unique? 2. Canyouseparatelyusethequotientruleandthepowerrule combinedwiththechainruletoderive d 1 ? dx cos2 x 3. Supposethatafunction y = f (x) satisfies x2 + (y − 1)2 = 1 forevery x ∈ [−1, 1]. Wheredoes f ′ (x) failtoexist? . QingwenHu (UTD) Lectures17-19 . . . . October, 2012 . 4/4