Exercises

In the derivation of our formulas so far it has been necessary to apply the General Rule, §4.7, (i.e. the four steps), only for the following:

III	$\frac{d}{dx}(u + v - w) = \frac{du}{dx}\ +\ \frac{dv}{dx}\ -\ \frac{dw}{dx}$	Algebraic sum.
V	$\frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx}.$	Product.
VII	$\frac{d}{dx} \left ( \frac{u}{v} \right ) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}.$	Quotient.
VIII	$\frac{d}{dx}(\log_a v) = \log_a e \frac{\frac{dv}{dx}}{v}.$	Logarithm.
XI	$\frac{d}{dx} (\sin\ v) = \cos\ v \frac{dv}{dx}$	Sine.
XXV	$\frac{dy}{dx} = \frac{dy}{dv} \cdot \frac{dv}{dx}.$	Function of a function.
XXVI	$\frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}.$	Inverse functions.

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.,

$$\lim_{v \to 0} \frac{\sin v}{1} = 1\ \ \ \ \ \ {\rm by\ \S \ref{sec:22}},% § 22, p. 21$$

and

$$\lim_{v \to 0}(1 + v)^{\frac{1}{v}} = e\ \ \ \ \ \ \ {\rm by\ \S \ref{sec:23}}. %By § 23, p. 22$$

Examples/exercises:

Differentiate the following:

1. $y = \sin (ax^2)$ .
   $$\frac{dy}{dx} = \cos ax^2 \frac{d}{dx}(ax^2),\ \ \ {\rm by\ XI\ } (v = ax^2).$$

2. $y = \tan \sqrt{1 - x}$.
   $$\begin{array}{ll} \frac{dy}{dx} & = \sec^2 \sqrt{1 - x} \fra... ...\\ & = -\frac{\sec^2 \sqrt{1 - x}}{2\sqrt{1 - x}}. \end{array}$$

3. $y = \cos^3x$.

   This may also be written, $y = (\cos x)^3$.

   $$\begin{array}{ll} \frac{dy}{dx} & = 3(\cos x)^2 \frac{d}{dx... ...sin x) \ \ \ {\rm by\ XII}\\ & = - 3\sin x\cos^2x. \end{array}$$

4. $y = \sin nx\sin^nx$.
   $$\begin{array}{ll} \frac{dy}{dx} & = \sin nx \frac{d}{dx}(\s... ... \cos nx\ sinx)\\ & = n\sin^{n - 1}x \sin(n + 1)x. \end{array}$$

5. $y = \sec ax$                                                Ans: $\frac{dy}{dx} = a\sec ax\tan ax$

6. $y = \tan(ax + b)$                                                Ans: $\frac{dy}{dx} = a\sec^2(ax + b)$

7. $s = \cos 3ax$                                                Ans: $\frac{ds}{dx} = - 3a\sin 3ax$

8. $s = \cot(2t^2 + 3)$                                                Ans: $\frac{ds}{dt} = - 4t\csc^2(2t^2 + 3)$

9. $f(y) = \sin 2y\cos y$                                Ans: $f'(y) = 2\cos 2y\cos y - \sin 2y\sin y$

10. $F(x) = \cot^2 5x$                                                Ans: $F'(x) = - 10\cot 5x\csc^2 5x$

11. $F(\theta ) = \tan \theta - \theta$                                                Ans: $F'(\theta) = \tan^2 \theta$

12. $f(φ) = \phi \sin\phi + \cos\phi$                                                Ans: $f'(\phi ) = \phi \cos \phi$

13. $f(t) = \sin^3 t\cos t$                                Ans: $f'(t) = \sin^2 t(3\cos t - \sin^2 t)$

14. $r = a\cos 2\theta$                                                Ans: $\frac{dr}{d\theta} = - 2a\sin 2\theta$

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

31. $y = \log \sqrt{ \frac{1 + \sin x}{1 - \sin x} }$                                                 Ans: $\frac{dy}{dx} = \frac{1}{\cos x}$

32. $y = \log \tan \left ( \frac{\pi}{4} + \frac{x}{2} \right )$                                                Ans: $\frac{dy}{dx} = \frac{1}{\cos x}$

33. $f(x) = \sin(x + a)\cos(x - a)$                                                Ans: $f'(x) = \cos 2x$

34. $y = a^{\tan nx}$                                                Ans: $y' = na^{\tan nx}\sec^2 nx\log a$

35. $y = e^{\cos x}\sin x$                                                Ans: $y' = e^{\cos x} (\cos x - \sin^2 x)$

36. $y = e^x\log\sin x$                                                Ans: $y'= e^x(\cot x + \log\sin x)$

37. Compute the following derivatives:
    $$\begin{array}{lll} (a) \ \ \frac{d}{dx} \sin 5x^2 & (f) \ \ ... ...x) & (o) \ \ \frac{d}{ds} \log \sqrt{1 - 2\sin^2 s} \end{array}$$

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

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

40. $f(\theta) = \frac{1 + \cos \theta}{1 - \cos \theta}$                                                Ans: $f'(\theta) =\ -\frac{2 \sin \theta}{(1 - \cos \theta)^2}$

41. $f(\phi) = \frac{e^{a\phi}(a \sin \phi - \cos \phi)}{a^2 + 1}$                                                Ans: $f'(\phi) =\ e^{a\phi} \sin \phi$

42. $f(s) = (s \cot s)^2$                                Ans: $f'(s) =\ 2s \cot s (\cot s - s \csc^2 s)$

43. $r = \frac{1}{3} \tan^3 \theta - \tan \theta + \theta$                                                Ans: $\frac{dr}{d\theta} = \tan^4\theta$

44. $y = x^{\sin x}$                                                Ans: $\frac{dy}{dx} =\ x^{\sin x} \left ( \frac{\sin x}{x} + \log x \cos x \right )$

45. $y = (\sin x)^x$                                                Ans: $y' =\ (\sin x)^x [ \log \sin x + x \cot x]$

46. $y = (\sin x)^{\tan x}$                                Ans: $y' = (\sin x)^{\tan x} (1 + \sec^2 x \log \sin x)$

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 $\frac{\cos v}{\sin v}$.