The Chemistry Maths Book, Second Edition

110 Chapter 4Differentiation EXAMPLE 4.18Inverse hyperbolic functions If thenx 1 = 1 asinh 1 yand Because cosh y 1 > 10 (see ...

4.8 Logarithmic differentiation 111 Solving for gives This is the result obtained in Example 4.6 by differentiation of the inver ...

112 Chapter 4Differentiation and, multiplying by y, (ii)y 1 = 1 a x ,ln 1 y 1 = 1 x 1 ln 1 a. Then (iii)y 1 = 1 x x ,ln 1 y 1 = ...

4.9 Successive differentiation 113 4.9 Successive differentiation The derivative of a function can itself be differentiated if i ...

114 Chapter 4Differentiation For example, n 1 = 1 3: f (3) (x) 1 = 1 f′′′(x) 1 = 1 −a 3 1 cos 1 ax n 1 = 1 4: f (4) (x) 1 = 1 (− ...

4.10 Stationary points 115 gradient decreases from positive values, through zero at A, to negative values. It follows that the r ...

116 Chapter 4Differentiation This is a maximum or minimum, depending on the sign of the fourth derivative. For example, the func ...

4.10 Stationary points 117 Division by 2βand multiplication by(1 1 − 1 c 2 ) 122 gives Thenε 1 = 1 α 1 ± 1 β. EXAMPLE 4.25Snell’ ...

118 Chapter 4Differentiation in which y 1 ,y 2 , andX 1 = 1 x 1 1 + 1 x 2 are constant. Then = 1 0 for a minimum Hence Snell’s l ...

4.12 The differential 119 Derivatives with respect to time are sometimes written in a ‘dot notation’: 6 (4.27) 0 Exercise 87 Ang ...

120 Chapter 4Differentiation Consider the cubic Let ∆ybe the change in yaccompanying the change ∆xin x: ∆y 1 = 1 f(x 1 + 1 ∆x) 1 ...

4.12 The differential 121 If the radius is increased by amount∆r, the corresponding change in the area is ∆A 1 = 1 π(r 1 + 1 ∆r) ...

122 Chapter 4Differentiation 8 In fact, developments in mathematical logic between 1920 and 1960 have led to the development of ...

Section 4.5 Differentiate from first principles: 2 x 2 1 + 13 x 1 + 14 19.x 4 22 x 2 21.x 322 22 ...

124 Chapter 4Differentiation 58.p(V 1 − 1 nb) 1 − 1 nRT 1 = 10 59. Differentiate 60.sin − 1 12 x 61.tan − 1 1 x 2 63.sinh − ...

4.13 Exercises 125 82.Confirm that the cubicy 1 = 1 x 3 1 − 17 x 2 1 + 116 x 1 − 110 , discussed in Example 2.23, has local maxi ...

5 Integration 5.1 Concepts Consider a body moving along a curve from point Aat timet 1 = 1 t A to point Bat time t 1 = 1 t B . L ...

5.2 The indefinite integral 127 It is readily verified that this distance is equal to the area, shaded in the figure, bounded by ...

128 Chapter 5Integration (5.2) where Cis an arbitrary constant. For example, if y 1 = 1 x 2 then 1 = 12 x, and the indefinite in ...

5.2 The indefinite integral 129 EXAMPLES 5.1Indefinite integrals (i) (ii) (iii) (iv) (v) (vi) We note that if we putC 1 = 1 ln 1 ...