By Brian H. Chirgwin, Charles Plumpton

ISBN-10: 0080063888

ISBN-13: 9780080063881

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**Example text**

I) If x = at2, y = 2 a t , then- (ii) If x = dy (-bcott) a cos t , y = b sin t , then - = dx a (iii) If z = a(O - sin 0), y = a(1 - cos 0 ) , then- dY = cot(+0 ) . dx Exercises 2:2 Differentiate the following functions w. r. to x: 1. x ( l - x2) 2. x ( l - 2) 6. x4 (1 - x) 7. x4(l - x ) ~ 8. x m ( l - 2") 11. (x 15. x"(2x - 12. x2 (x &)8 X2 19. 1 -- x2 X 23. _ _ 1 - 22 28. + 3x2 2 1 - 4x2 ~ - 41. sin x ~ 1 + sin x 1 - sin 2 s 45. 1 sin 22 + 50. x2 sec x 13. (x + &r X (2" 24. ~ X 1 - 3x2 a ~ + bx" c - dxm 25.

Hence find the greatest possible value of (i) x, (ii) y. 16. Show t h a t in general two circles of the (coaxal) system x2 + y2 + 2λχ — a2 = 0, where λ is a variable parameter, can be drawn to touch an arbitrary straight line, and t h a t if the two circles cut orthogonally, t h e rectangle contained b y the per pendiculars drawn to the straight line from t h e common points of the system is equal to a2. Ex. I INTRODUCTORY CONCEPTS 31 17. Show that the equation ax2 + 2Àxy — ay2 = 0 represents two perpendicular straight lines.

R . t o x using t h e chain rule where necessary for differentiating t e r m s involving y. This process m a y be repeated t o obtain second a n d higher derivatives. Examples, (i) If log(x2 + y2) = 2 tan _ 1 (y/#), find dy/dx and d2y/dx2 in terms of x and y. Differentiation gives dy dy Λ( 2x + 2y 2{x y dx dx 2 2 2 2 x -f y x + y dy dx . d 2y ' dx2 (x- + y y - »> i 1 +-sì) - <* "* 2y (x — y)2 (x — y)2 2y (x — y)2 ( 2x(x + y) (x — yf +- 2(x2 + y2) (x — yf (ii) If x eV = cos y, find dy/dx and d2y/dx2 when x = 1, y xey—— = dx Putting x = l, y dy 2x dx (x - y) dy - sin y dx ■0.

### A Course of Mathematics for Engineers and Scientists. Volume 1 by Brian H. Chirgwin, Charles Plumpton

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