In these examples $m$ is a rational number and $a,b,\dots,\alpha,\beta,\dots$ have such values that the functions which involve them are real.
1. Find the derivatives of
\[\begin{gathered} \cos^{m} x, \quad \sin^{m} x, \quad \cos x^{m}, \quad \sin x^{m}, \quad \cos (\sin x), \quad \sin (\cos x),\\ \sqrt{a^{2}\cos^{2} x + b^{2}\sin^{2} x}, \quad \frac{\cos x\sin x}{\sqrt{a^{2}\cos^{2} x + b^{2}\sin^{2} x}},\\ x\arcsin x + \sqrt{1 – x^{2}}, \quad (1 + x)\arctan\sqrt{x} – \sqrt{x}.\end{gathered}\]
2. Verify by differentiation that \(\arcsin x + \arccos x\) is constant for all values of \(x\) between \(0\) and \(1\), and \(\arctan x + \operatorname{arccot} x\) for all positive values of \(x\).
3. Find the derivatives of \[\arcsin\sqrt{1 – x^{2}},\quad \arcsin\{2x\sqrt{1 – x^{2}}\},\quad \arctan \left(\frac{a + x}{1 – ax}\right).\] How do you explain the simplicity of the results?
4. Differentiate \[\frac{1}{\sqrt{ac – b^{2}}} \arctan \frac{ax + b}{\sqrt{ac – b^{2}}},\quad -\frac{1}{\sqrt{-a}} \arcsin\frac{ax + b}{\sqrt{b^{2} – ac}}.\]
5. Show that each of the functions \[2\arcsin \sqrt{\frac{x – \beta}{\alpha – \beta}},\quad 2\arctan \sqrt{\frac{x – \beta}{\alpha – x}},\quad \arcsin \frac{2\sqrt{(\alpha – x)(x – \beta)}}{\alpha – \beta}\] has the derivative \[\frac{1}{\sqrt{(\alpha – x)(x – \beta)}}.\]
6. Prove that \[\frac{d}{d\theta}\left\{ \arccos \sqrt{\frac{\cos 3\theta}{\cos^{3}\theta}} \right\} = \sqrt{\frac{3}{\cos\theta \cos 3\theta}}.\]
7. Show that \[\frac{1}{\sqrt{C(Ac – aC)}}\, \frac{d}{dx} \left[ \arccos \sqrt{\frac{C(ax^{2} + c)}{c(Ax^{2} + C)}} \right] = \frac{1}{(Ax^{2} + C) \sqrt{ax^{2} + c}}.\]
8. Each of the functions \[\frac{1}{\sqrt{a^{2} – b^{2}}} \arccos \left(\frac{a\cos x + b}{a + b\cos x}\right),\quad \frac{2}{\sqrt{a^{2} – b^{2}}} \arctan \left\{\sqrt{\frac{a – b}{a + b }} \tan \tfrac{1}{2}x\right\}\] has the derivative \(1/(a + b\cos x)\).
9. If \(X = a + b\cos x + c\sin x\), and \[y = \frac{1}{\sqrt{a^{2} – b^{2} -c^{2}}} \arccos \frac{aX – a^{2} + b^{2} + c^{2}}{X \sqrt{b^{2} + c^{2}}},\] then \(dy/dx = 1/X\).
10. Prove that the derivative of \(F[f\{\phi(x)\}]\) is \(F'[f\{\phi(x)\}]\, f’\{\phi(x)\}\phi'(x)\), and extend the result to still more complicated cases.
11. If \(u\) and \(v\) are functions of \(x\), then \[D_{x} \arctan(u/v) = (vD_{x}u – uD_{x}v)/(u^{2} + v^{2}).\]
12. The derivative of \(y = (\tan x + \sec x)^{m}\) is \(my\sec x\).
13. The derivative of \(y = \cos x + i\sin x\) is \(iy\).
14. Differentiate \(x\cos x\), \((\sin x)/x\). Show that the values of \(x\) for which the tangents to the curves \(y = x\cos x\), \(y = (\sin x)/x\) are parallel to the axis of \(x\) are roots of \(\cot x = x\), \(\tan x = x\) respectively.
15. It is easy to see (cf. EX. XVII. 5) that the equation \(\sin x = ax\), where \(a\) is positive, has no real roots except \(x = 0\) if \(a \geq 1\), and if \(a < 1\) a finite number of roots which increases as \(a\) diminishes. Prove that the values of \(a\) for which the number of roots changes are the values of \(\cos\xi\), where \(\xi\) is a positive root of the equation \(\tan\xi = \xi\). [The values required are the values of \(a\) for which \(y = ax\) touches \(y = \sin x\).]
16. If \(\phi(x) = x^{2}\sin(1/x)\) when \(x \neq 0\), and \(\phi(0) = 0\), then \[\phi'(x) = 2x\sin(1/x) – \cos(1/x)\] when \(x\neq 0\), and \(\phi'(0) = 0\). And \(\phi'(x)\) is discontinuous for \(x = 0\) (cf. § 111, (2)).
17. Find the equations of the tangent and normal at the point \((x_{0}, y_{0})\) of the circle \(x^{2} + y^{2} = a^{2}\).
[Here
\(y = \sqrt{a^{2} – x^{2}}\),
\(dy/dx = -x/\sqrt{a^{2} – x^{2}}\), and the tangent is
\[y – y_{0} = (x – x_{0}) \left\{-x_{0}/\sqrt{a^{2} – x_{0}^{2}}\right\},\] which may be reduced to the form
\(xx_{0} + yy_{0} = a^{2}\). The normal is
\(xy_{0} – yx_{0} = 0\), which of course passes through the origin.]
18. Find the equations of the tangent and normal at any point of the ellipse \((x/a)^{2} + (y/b)^{2} = 1\) and the hyperbola \((x/a)^{2} – (y/b)^{2} = 1\).
19. The equations of the tangent and normal to the curve \(x = \phi(t)\), \(y = \psi(t)\), at the point whose parameter is \(t\), are \[\frac{x – \phi(t)}{\phi'(t)} = \frac{y – \psi(t)}{\psi'(t)},\quad \{x – \phi(t)\} \phi'(t) + \{y – \psi(t)\} \psi'(t) = 0.\]
