Differentiate the functions given in Exercises 1 to 11 w.r.t. x. 1. cos x . cos 2x . cos 3x 2. ( 1) ( 2) ( 3) ( 4) ( 5) x x x x x 3. (log x)cos x 4. xx 2sin x 5. (x + 3)2 . (x + 4)3 . (x + 5)4 6. 1 1 1 x x x x x + + + 7. (log x)x + xlog x 8. (sin x)x + sin1 x 9. xsin x + (sin x)cos x 10. 2 cos 2 1 1 x x x x x + + 11. (x cos x)x + 1 (xsin x) x Find dy dx of the functions given in Exercises 12 to 15. 12. xy + yx = 1 13. yx = xy 14. (cos x)y = (cos y)x 15. xy = e(x y) 16. Find the derivative of the function given by f (x) = (1 + x) (1 + x2) (1 + x4) (1 + x8) and hence find f (1). 17. Differentiate (x2 5x + 8) (x3 + 7x + 9) in three ways mentioned below: (i) by using product rule (ii) b
Differentiate the functions given in Exercises 1 to 11 w.r.t. x.
Question 1:cos x . cos 2x . cos 3x
Question 2: ( 1) ( 2) ( 3) ( 4) ( 5) x x x x x − − − − −
Question 3:(log x)cos x
Question 4:xx – 2sin x
Question 5:(x + 3)2 . (x + 4)3 . (x + 5)4
Question 6: 1 1 1 x x x x x + + +
Question 7:(log x)x + xlog x
Question 8:(sin x)x + sin–1 x
Question 9:xsin x + (sin x)cos x
Question 10: 2 cos 2 1 1 x x x x x + + −
Question 11:(x cos x)x + 1 (xsin x) x Find dy dx of the functions given in Exercises 12 to 15
Question 12:xy + yx = 1
Question 13:yx = xy
Question 14:(cos x)y = (cos y)x
Question 15:xy = e(x – y)
Question 16:Find the derivative of the function given by f (x) = (1 + x) (1 + x2) (1 + x4) (1 + x8) and hence find f ¢(1).
Question 17: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?
Question 18: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.
