>	F:=(x,y)->log(1+x^4+y^4)/sqrt(x^2+y^2);

$F:=\lambda \left(x,y,\frac{\ln \left(1+x^4+y^4\right)}{\sqrt{x^2+y^2}}\right)$

>	Diff(F(x,y),x,x):%=value(%);

$\frac{{\partial }^2}{\partial x^2}\left(\frac{\ln \left(1+x^4+y^4\right)}{\sqrt{\left(x^2+y^2\right)}}\right)=12\frac{x^2}{\left(1+x^4+y^4\right)\sqrt{\left(x^2+y^2\right)}}-16\frac{x^6}{{\left(1+x^4+y^4\right)}^2\sqrt{\left(x^2+y^2\right)}}-8\frac{x^4}{\left(1+x^4+y^4\right){\left(x^2+y^2\right)}^{\frac{3}{2}}}+3\frac{\ln \left(1+x^4+y^4\right)x^2}{{\left(x^2+y^2\right)}^{\frac{5}{2}}}-\frac{\ln \left(1+x^4+y^4\right)}{{\left(x^2+y^2\right)}^{\frac{3}{2}}}$

>	Diff(F(x,y),x,y):%=value(%);

$\frac{{\partial }^2}{\partial x\partial y}\left(\frac{\ln \left(1+x^4+y^4\right)}{\sqrt{\left(x^2+y^2\right)}}\right)=-16\frac{x^3{y}^3}{{\left(1+x^4+y^4\right)}^2\sqrt{\left(x^2+y^2\right)}}-4\frac{x^{3}y}{\left(1+x^4+y^4\right){\left(x^2+y^2\right)}^{\frac{3}{2}}}-4\frac{y^{3}x}{\left(1+x^4+y^4\right){\left(x^2+y^2\right)}^{\frac{3}{2}}}+3\frac{\ln \left(1+x^4+y^4\right)xy}{{\left(x^2+y^2\right)}^{\frac{5}{2}}}$

>	Diff(F(x,y),y,x):%=value(%);

$\frac{{\partial }^2}{\partial x\partial y}\left(\frac{\ln \left(1+x^4+y^4\right)}{\sqrt{\left(x^2+y^2\right)}}\right)=-16\frac{x^3{y}^3}{{\left(1+x^4+y^4\right)}^2\sqrt{\left(x^2+y^2\right)}}-4\frac{x^{3}y}{\left(1+x^4+y^4\right){\left(x^2+y^2\right)}^{\frac{3}{2}}}-4\frac{y^{3}x}{\left(1+x^4+y^4\right){\left(x^2+y^2\right)}^{\frac{3}{2}}}+3\frac{\ln \left(1+x^4+y^4\right)xy}{{\left(x^2+y^2\right)}^{\frac{5}{2}}}$

>	Diff(F(x,y),y,y):%=value(%);

$\frac{{\partial }^2}{\partial y^2}\left(\frac{\ln \left(1+x^4+y^4\right)}{\sqrt{\left(x^2+y^2\right)}}\right)=12\frac{y^2}{\left(1+x^4+y^4\right)\sqrt{\left(x^2+y^2\right)}}-16\frac{y^6}{{\left(1+x^4+y^4\right)}^2\sqrt{\left(x^2+y^2\right)}}-8\frac{y^4}{\left(1+x^4+y^4\right){\left(x^2+y^2\right)}^{\frac{3}{2}}}+3\frac{\ln \left(1+x^4+y^4\right)y^2}{{\left(x^2+y^2\right)}^{\frac{5}{2}}}-\frac{\ln \left(1+x^4+y^4\right)}{{\left(x^2+y^2\right)}^{\frac{3}{2}}}$

2)

>	alias(y=y(x)):

>	G:=sqrt(x)+sqrt(y)=1;

$G:=\sqrt{x}+\sqrt{y}=1$

>	c:=diff(G,x);

$c:=\frac{1}{2\sqrt{x}}+\frac{1}{2}\frac{\frac{d}{dx}y}{\sqrt{y}}=0$

>	d:=solve(c, diff(y,x));

$d:=-\frac{\sqrt{y}}{\sqrt{x}}$

>	a:=diff(G,x$2);

$a:=-\frac{1}{4x^{\frac{3}{2}}}-\frac{1}{4}\frac{{\left(\frac{d}{dx}y\right)}^2}{y^{\frac{3}{2}}}+\frac{1}{2}\frac{\frac{d^2}{{dx}^2}y}{\sqrt{y}}=0$

>	b:=solve(a, diff(y,x$2));

$b:=-\frac{1}{2}\frac{-y^{\frac{3}{2}}-{\left(\frac{d}{dx}y\right)}^2{x}^{\frac{3}{2}}}{x^{\frac{3}{2}}y}$

>	subs(diff(y,x)=d,b);

$-\frac{1}{2}\frac{-y^{\frac{3}{2}}-y\sqrt{x}}{x^{\frac{3}{2}}y}$

>

restart;

3)

>	a:int(sqrt(exp(x)-1),x);

$2\sqrt{\left(e^x-1\right)}-2\arctan \left(\sqrt{\left(e^x-1\right)}\right)$

>	normal(diff(%,x));

$\sqrt{\left(e^x-1\right)}$

>

VypocetIntegralu := proc (vyraz) local r, h, cesky, preklad, simple, vypis; cesky := false; vypis := false; if nargs = 2 and args[2] = czech then cesky := true end if; r := Int(vyraz,x); print(r = ``);  while h <> [] do h := Student :-Calculus1:-Hint(r); if 1 < nops({h}) then h := op(1,{h}) end if; preklad := h; if h <> [] then vypis := true; r := Student:-Calculus1:-Rule[h](r); if cesky then if h[1] = rewrite then preklad[1] := `upravíme výraz` end if; if h[ 1] = sum then preklad[1] := `rozepíšeme součet` end if; if h[1] = constant then preklad[1] := `integrujeme konstantu` end if; if h[1] = constantmultiple then preklad[1] := `vytkneme konstantu`; vypis := false end if; if h[1] = partialfractions then preklad[1] := `rozložíme na parciální zlomky` end if; if h[1] = change then preklad := [`zavedeme substituci`, h[2]] end if; if h[1] = power or h[1] = sin or h[1] = cos or h[1] = exp then preklad[1] := `integrujeme výraz` end if; if h[1] = revert then preklad[1] := `dosadíme zpět` end if; if h[1] = parts then preklad := [ `užijeme metodu per partes`, u = h[2], v = h[3]] end if; if h[1] = solve then preklad[1] := `integrál spočítáme algebraicky` end if end if; if vypis then print(preklad); print(`` = rhs(r)) end if end if end do; if vypis then simple := simplify(rhs(r)); if simple <> rhs(r) then print([`upravíme výraz`]); print (`` = simple) end if end if end proc:

>	VypocetIntegralu(sqrt(exp(x)-1),czech);

$\int \sqrt{\left(e^x-1\right)}dx=$

$\left[\mathrm{zavedeme\; substituci},u=e^x\right]$

$=\int \frac{\sqrt{\left(u-1\right)}}{u}du$

$\left[\mathrm{zavedeme\; substituci},u-1={\mathrm{u1}}^2\right]$

$=\int \left(2-\frac{2}{1+{\mathrm{u1}}^2}\right)d\mathrm{u1}$

$\left[\mathrm{rozepíaeme\; sou\; et}\right]$

$=\int 2d\mathrm{u1}+\int -\frac{2}{1+{\mathrm{u1}}^2}d\mathrm{u1}$

$\left[\mathrm{integrujeme\; konstantu}\right]$

$=2\mathrm{u1}+\int -\frac{2}{1+{\mathrm{u1}}^2}d\mathrm{u1}$

$\left[\mathrm{zavedeme\; substituci},\mathrm{u1}=\tan \left(\mathrm{u2}\right)\right]$

$=2\mathrm{u1}-2\int 1d\mathrm{u2}$

$\left[\mathrm{integrujeme\; konstantu}\right]$

$=2\mathrm{u1}-2\mathrm{u2}$

$=2\mathrm{u1}-2\arctan \left(\mathrm{u1}\right)$

$=2\sqrt{\left(u-1\right)}-2\arctan \left(\sqrt{\left(u-1\right)}\right)$

$=2\sqrt{\left(e^x-1\right)}-2\arctan \left(\sqrt{\left(e^x-1\right)}\right)$

4)

>	int(1/sqrt(1-x^2),x=0..1);

$\frac{1}{2}\mathrm{\pi }$

5)

>	sum((k^2+k-1)/(k+2)!,k=1..infinity);

$\frac{1}{2}$

6)

>	limit(log(x)/exp(x),x=infinity);

$0$

7)

>	B:=taylor(sin(x), x=0,6);

$B:=x-\frac{1}{6}{x}^3+\frac{1}{120}{x}^5+O\left(x^6\right)$

>	F:=convert(B, 'polynom');

>	f:=unapply(F,x);

$f:=\lambda \left(x,x-\frac{1}{6}{x}^3+\frac{1}{120}{x}^5\right)$

>	with(plots):

>	plot([sin(x),f(x)], x=-Pi..Pi);

>