Elements of the Differential and Integral Calculus/Chapter V part 3

54. Differentiation of $$\operatorname{vers} \ v$$.

Not only do all the other formulas we have deduced depend on these, but all we shall deduce hereafter depend on them as well. Hence it follows that the derivation of the fundamental formulas for differentiation involves the calculation of only two limits of any difficulty, viz.,

EXAMPLES

Differentiate the following:

15. $$\frac{d}{dx} \sin^2 x = \sin 2x$$.

16. $$\frac{d}{dx} \cos^3 x^2 = -6x \cos^2 x^2 \sin x^2$$.

17. $$\frac{d}{dt} \csc \frac{t^2}{2} = -t \csc \frac{t^2}{2} \cot \frac{t^2}{2}$$.

18. $$\frac{d}{ds} a \sqrt{\cos 2s} = -\frac{a \sin 2s}{\sqrt{\cos 2s}}$$.

19. $$\frac{d}{d\theta} a(1 - \cos \theta) = a sin \theta$$.

20. $$\frac{d}{dx}(\log \cos x) = -\tan x$$.

21. $$\frac{d}{dx}(\log \tan x) = \frac{2}{\sin 2x}$$.

22. $$\frac{d}{dx}(\log \sin^2 x) = 2 \cot x$$.

23. $$\frac{d}{dt} \cos \frac{a}{t} = \frac{a}{t^2} \sin \frac{a}{t}$$.

24. $$\frac{d}{d\theta} \sin \frac{1}{\theta^2} = -\frac{2}{\theta^3} \cos \frac{1}{\theta^2}$$.

25. $$\frac{d}{dx} e^{\sin x} = e^{\sin x} \cos x$$.

26. $$\frac{d}{dx} \sin(\log x) = \frac{\cos(\log x)}{x}$$.

27. $$\frac{d}{dx} \tan(\log x) = \frac{\sec^2(\log x)}{x}$$.

28. $$\frac{d}{dx} a \sin^3 \frac{\theta}{3} = a \sin^2 \frac{\theta}{3} \cos \frac{\theta}{3}$$.

29. $$\frac{d}{d\alpha} \sin(\cos \alpha) = -\sin \alpha \cos(\cos \alpha)$$.

30. $$\frac{d}{dx} \frac{\tan x - 1}{\sec x} = \sin x + \cos x$$.

37. Differentiate the following functions:

38. $$\frac{d}{dx}(x^n e^{\sin x}) = x^{n - 1} e^{\sin x} (n + x\cos x)$$.

39. $$\frac{d}{dx} (e^{ax} \cos mx) = e^{ax}(a \cos mx - m \sin mx)$$.

47. Prove $$\frac{d}{dx} \cos v = -\sin v \frac{dv}{dx}$$, using the General Rule.

48. Prove $$\frac{d}{dx} \cot v = -\csc^2 v \frac{dv}{dx}$$ by replacing $$\cot v$$ by $$\frac{\cos c}{\sin v}$$.

55. Differentiation of $$\arcsin v$$.

56. Differentiation of $$\arccos v$$.

57. Differentiation of $$\arctan v$$.

58. Differentiation of $$\arccot u$$.

Following the method of the last section, we get

XXI $$\frac{d}{dx}(\arccot v) = -\frac{\frac{dv}{dx}}{1 + v^2}$$.

59. Differentiation of $$\arcsec u$$.

60. Differentiation of $$\arccsc v$$.

61. Differentiation of $$\operatorname{arc vers} v$$.