Higher Engineering Mathematics

322 DIFFERENTIAL CALCULUS x^2 +y^2 =25 is the equation of a circle, centre at the origin and radius 5, as shown in Fig. 30.1. At ...

DIFFERENTIATION OF IMPLICIT FUNCTIONS 323 G Gradient (^0124) x Gradient r^ = √ 5 − 1 − 2 3 2 1 y 4 x^2 +y^2 − 2 x− 2 y= 3 =^12 = ...

Differential calculus 31 Logarithmic differentiation 31.1 Introduction to logarithmic differentiation With certain functions con ...

LOGARITHMIC DIFFERENTIATION 325 G (ii) Apply the laws of logarithms. Thus lny=ln(1+x)^2 +ln(x−1) 1 2 −lnx−ln(x+2) 1 (^2) ,bylaws ...

326 DIFFERENTIAL CALCULUS Whenx=3, dy dx = √ (1)^3 (4)^2 (5) ( 3 2 − 2 4 − 2 5 ) =± 1 80 ( 3 5 ) =± 3 400 or± 0. 0075 Problem 3. ...

LOGARITHMIC DIFFERENTIATION 327 G 4.y= e^2 xcos 3x √ (x−4) [ e^2 xcos 3x √ (x−4) { 2 −3 tan 3x− 1 2(x−4) }] 5.y= 3 θsinθcosθ [ 3 ...

328 DIFFERENTIAL CALCULUS i.e. dy dx =x √ (x− 1 ) { 1 x(x− 1 ) − ln(x− 1 ) x^2 } (b) Whenx=2, dy dx =^2 √ (1) { 1 2(1) − ln (1) ...

Assign-08-H8152.tex 23/6/2006 15: 10 Page 329 G Differential calculus Assignment 8 This assignment covers the material contained ...

Differential calculus 32 Differentiation of hyperbolic functions 32.1 Standard differential coefficients of hyperbolic functions ...

DIFFERENTIATION OF HYPERBOLIC FUNCTIONS 331 G 32.2 Further worked problems on differentiation of hyperbolic functions Problem 3. ...

Differential calculus 33 Differentiation of inverse trigonometric and hyperbolic functions 33.1 Inverse functions Ify= 3 x−2, th ...

DIFFERENTIATION OF INVERSE TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS 333 G y 3 π/2 π/2 π −π/2 −π − 3 π/2 + 1 x B y = sin^1 x A − 1 ...

334 DIFFERENTIAL CALCULUS Letu=f(x) theny=sin−^1 u Then du dx =f′(x) and dy du = 1 √ 1 −u^2 (see para. (i)) Thus dy dx = dy du × ...

DIFFERENTIATION OF INVERSE TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS 335 G Hence, when y=cos−^1 (1− 2 x^2 ) then dy dx = −(− 4 x) √ ...

336 DIFFERENTIAL CALCULUS Problem 7. Differentiatey=xcosec−^1 x. Using the product rule: dy dx =(x) [ − 1 x √ x^2 − 1 ] +(cosec− ...

DIFFERENTIATION OF INVERSE TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS 337 G (a)θ^2 cos−^1 (θ^2 −1) (b) (1−x^2 ) tan−^1 x ⎡ ⎢ ⎢ ⎣ ( ...

338 DIFFERENTIAL CALCULUS Thus, x a = e^2 y− 1 e^2 y+ 1 from which, x(e^2 y+1)=a(e^2 y−1) Hencex+a=ae^2 y−xe^2 y=e^2 y(a−x) from ...

DIFFERENTIATION OF INVERSE TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS 339 G [An alternative method of differentiating sinh−^1 x a is ...

340 DIFFERENTIAL CALCULUS Problem 14. Show that d dx [ tanh−^1 x a ] = a a^2 −x^2 and hence determine the differential coefficie ...

DIFFERENTIATION OF INVERSE TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS 341 G Problem 19. Determine ∫ dx √ (x^2 +4) Since d dx ( sinh− ...