119. Differentiation of transcendental functions

1. Find the derivatives of

$$\begin{array}{c}{\cos }^{m}x,{\sin }^{m}x,\cos x^m,\sin x^m,\cos (\sin x),\sin (\cos x),\\ \sqrt{a^2{\cos }^{2}x+b^2{\sin }^{2}x},\frac{\cos x\sin x}{\sqrt{a^2{\cos }^{2}x+b^2{\sin }^{2}x}},\\ x\arcsin x+\sqrt{1–x^2},(1+x)\arctan \sqrt{x}–\sqrt{x}.\end{array}$$

2. Verify by differentiation that $\arcsin x+\arccos x$ is constant for all values of $x$ between $0$ and $1$, and $\arctan x+\mathrm{arccot}x$ for all positive values of $x$.

3. Find the derivatives of $$\arcsin \sqrt{1–x^2},\arcsin \{2x\sqrt{1–x^2}\},\arctan \left(\frac{a+x}{1–ax}\right).$$ How do you explain the simplicity of the results?

4. Differentiate $$\frac{1}{\sqrt{ac–b^2}}\arctan \frac{ax+b}{\sqrt{ac–b^2}},-\frac{1}{\sqrt{-a}}\arcsin \frac{ax+b}{\sqrt{b^2–ac}}.$$

5. Show that each of the functions $$2\arcsin \sqrt{\frac{x–\beta }{\alpha –\beta }},2\arctan \sqrt{\frac{x–\beta }{\alpha –x}},\arcsin \frac{2\sqrt{(\alpha –x)(x–\beta )}}{\alpha –\beta }$$ has the derivative $$\frac{1}{\sqrt{(\alpha –x)(x–\beta )}}.$$

6. Prove that $$\frac{d}{d\theta }\{\arccos \sqrt{\frac{\cos 3\theta }{{\cos }^3\theta }}\}=\sqrt{\frac{3}{\cos \theta \cos 3\theta }}.$$

7. Show that $$\frac{1}{\sqrt{C(Ac–aC)}}\frac{d}{dx}[\arccos \sqrt{\frac{C(a{x}^2+c)}{c(A{x}^2+C)}}]=\frac{1}{(A{x}^2+C)\sqrt{a{x}^2+c}}.$$

8. Each of the functions $$\frac{1}{\sqrt{a^2–b^2}}\arccos \left(\frac{a\cos x+b}{a+b\cos x}\right),\frac{2}{\sqrt{a^2–b^2}}\arctan \{\sqrt{\frac{a–b}{a+b}}\tan \frac{1}{2}x\}$$ 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 $F[f\{\varphi (x)\}]$ is $F^{\prime }[f\{\varphi (x)\}]f^{\prime }\{\varphi (x)\}{\varphi }^{\prime }(x)$, and extend the result to still more complicated cases.

11. If $u$ and $v$ are functions of $x$, then $$D_x\arctan (u/v)=(v{D}_{x}u–u{D}_{x}v)/(u^2+v^2).$$

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 $a\ge 1$, 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 $\varphi (x)=x^2\sin (1/x)$ when $x\ne 0$, and $\varphi (0)=0$, then $${\varphi }^{\prime }(x)=2x\sin (1/x)–\cos (1/x)$$ when $x\ne 0$, and ${\varphi }^{\prime }(0)=0$. And ${\varphi }^{\prime }(x)$ 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 $$y–y_0=(x–x_0)\{-x_0/\sqrt{a^2–x_0^2}\},$$ which may be reduced to the form $x{x}_0+y{y}_0=a^2$. The normal is $x{y}_0–y{x}_0=0$, 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 $x=\varphi (t)$, $y=\psi (t)$, at the point whose parameter is $t$, are $$\frac{x–\varphi (t)}{{\varphi }^{\prime }(t)}=\frac{y–\psi (t)}{{\psi }^{\prime }(t)},\{x–\varphi (t)\}{\varphi }^{\prime }(t)+\{y–\psi (t)\}{\psi }^{\prime }(t)=0.$$