Higher Engineering Mathematics, Sixth Edition
322 Higher Engineering Mathematics Thus d dx ( 3 y 2 x ) = ( 2 x) d dx ( 3 y)−( 3 y) d dx ( 2 x) ( 2 x)^2 = ( 2 x) ( 3 dy dx ) − ...
Differentiation of implicit functions 323 i.e. 2 x+ 2 y dy dx = 0 Hence dy dx =− 2 x 2 y =− x y Sincex^2 +y^2 =25, whenx=4,y= √ ...
324 Higher Engineering Mathematics Whenx=2andy=3, dy dx = 1 − 2 3 − 1 = − 1 2 Hence the gradients of the tangents are± 1 2 The c ...
Chapter 31 Logarithmic Differentiation 31.1 Introduction to logarithmic differentiation With certain functions containing more c ...
326 Higher Engineering Mathematics ln(cos 3x) [−3tan3x] ln( 3 x^3 +x) [ 9 x^2 + 1 3 x^3 +x ] ln( 5 x^2 + 10 x− 7 ) [ 10 x+ 1 ...
Logarithmic Differentiation 327 (v) Substituting forygives: dy dx = (x+ 1 )(x− 2 )^3 (x− 3 ) { 1 (x+ 1 ) + 3 (x− 2 ) − 1 (x− 3 ) ...
328 Higher Engineering Mathematics (iv) dy dx =y { 3 x + 1 xln2x − 1 −cotx } (v) dy dx = x^3 ln2x exsinx { 3 x + 1 xln2x − 1 −co ...
Logarithmic Differentiation 329 Differentiating both sides with respect toxgives: 1 y dy dx =(x) ( 1 x+ 2 ) +[ln(x+ 2 )]( 1 ), b ...
Revision Test This Revision Test covers the material contained in Chapters 27 to 31.The marks for each question are shown in bra ...
Chapter 32 Differentiation of hyperbolic functions 32.1 Standard differential coefficients of hyperbolic functions From Chapter ...
332 Higher Engineering Mathematics (b) d dθ (cothθ)= d dθ ( chθ shθ ) = (shθ)(shθ)−(chθ)(chθ) sh^2 θ = sh^2 θ−ch^2 θ sh^2 θ = −( ...
Differentiation of hyperbolic functions 333 Now try the following exercise Exercise 134 Further problemson differentiation of hy ...
Chapter 33 Differentiation of inverse trigonometric and hyperbolic functions 33.1 Inverse functions Ify= 3 x−2, then by transpos ...
Differentiation of inverse trigonometric and hyperbolic functions 335 (a) (d) (e) (f) (b) (c) y B A 21 0 11 x 3 /2 23 /2 /2 2 ...
336 Higher Engineering Mathematics Thus dy dx = 1 dx dy = 1 √ a^2 −x^2 i.e.when y=sin−^1 x a then dy dx = 1 √ a^2 −x^2 Since int ...
Differentiation of inverse trigonometric and hyperbolic functions 337 is negative betweenCandDand thus the differential coeffici ...
338 Higher Engineering Mathematics Problem 6. Differentiatey= cot−^12 x 1 + 4 x^2 Using the quotient rule: dy dx = ( 1 + 4 x^2 ) ...
Differentiation of inverse trigonometric and hyperbolic functions 339 Show that the differential coefficient of tan−^1 ( x 1 − ...
340 Higher Engineering Mathematics (iv) press ) to close the brackets (v) press=and 1.443635475 appears Hence,sinh−^12 = 1. 4436 ...
Differentiation of inverse trigonometric and hyperbolic functions 341 33.4 Differentiation of inverse hyperbolic functions Ify=s ...
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