D. Transcendental Functions. We have already proved (Ex. XXXIX. 4) that $D_{x} \sin x = \cos x, \quad D_{x} \cos x = -\sin x.$

By means of Theorems (4) and (5) of § 113, the reader will easily verify that \begin{aligned} {2} D_{x} \tan x &= \sec^{2} x, & D_{x} \cot x &= -\csc^{2} x,\\ D_{x} \sec x &= \tan x \sec x, \quad & D_{x} \csc x &= -\cot x\csc x.\end{aligned} And by means of Theorem (7) we can determine the derivatives of the ordinary inverse trigonometrical functions. The reader should verify the following formulae: \begin{aligned} {2} D_{x} \arcsin x &= \pm1/\sqrt{1 – x^{2}}, & D_{x} \arccos x &= \mp 1/\sqrt{1 – x^{2}},\\ D_{x} \arctan x &= 1/(1 + x^{2}), & D_{x} \operatorname{arccot} x &= -1/(1 + x^{2}),\\ D_{x} \operatorname{arcsec} x &= \pm 1/\{x\sqrt{x^{2} – 1}\}, \quad & D_{x} \operatorname{arccose}c x &= \mp 1/\{x\sqrt{x^{2} – 1}\}.\end{aligned} In the case of the inverse sine and cosecant the ambiguous sign is the same as that of $$\cos(\arcsin x)$$, in the case of the inverse cosine and secant the same as that of $$\sin(\arccos x)$$.

The more general formulae $D_{x} \arcsin(x/a) = \pm1/\sqrt{a^{2} – x^{2}},\quad D_{x} \arctan(x/a) = a/(x^{2} + a^{2}),$ which are also easily derived from Theorem (7) of § 113, are also of considerable importance. In the first of them the ambiguous sign is the same as that of $$a\cos\{\arcsin(x/a)\}$$, since $a\sqrt{1 – (x^{2}/a^{2})} = \pm\sqrt{a^{2} – x^{2}}$ according as $$a$$ is positive or negative.

Finally, by means of Theorem (6) of § 113, we are enabled to differentiate composite functions involving symbols both of algebraical and trigonometrical functionality, and so to write down the derivative of any such function as occurs in the following examples.

Example XLIV

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 $\frac{x – \phi(t)}{\phi'(t)} = \frac{y – \psi(t)}{\psi'(t)},\quad \{x – \phi(t)\} \phi'(t) + \{y – \psi(t)\} \psi'(t) = 0.$