Chapter 5
cos x . cos 2 x . cos 3 x 2. ( 1) ( 2) ( 3) ( 4) ( 5) / x x
| x | x | x |
|---|
− − / − − −
(log x)cos x 4. xx - 2 sin x
(x + 3)2 . (x + 4) 3 . ( x + 5) 4 6. x
| xx | x | x |
|---|
+ + +
(log x)x + xlog x 8. (sin x)x + sin-1 x
xsin x + (sin x)cos x 10. cos 2/1 x x xx x ++ −
(x cos x)x + ( sin ) xx x Find dy dx of the functions given in Exercises 12 to 15.
xy + yx = 1 13. yx = xy
(cos x)y = (cos y)x 15. xy = e(x - y)
Find the derivative of the function given by f (x) = (1 + x) (1 + x2) (1 + x4) (1 + x8) and hence find f ′(1).
Differentiate (x2 - 5x + 8) (x3 + 7x + 9) in three ways mentioned below: (i) by using product rule (ii) by expanding the product to obtain a single polynomial. (iii) by logarithmic differentiation. Do they all give the same answer?
If u, v and w are functions of x, then show that d dx (u. v. w) = du dx v. w + u . dv dx . w + u . v dw dx in two ways - first by repeated application of product rule, second by logarithmic differentiation.