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

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

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

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

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

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

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

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

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

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

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);

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

 > 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);

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

 > 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);

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

4)

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

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

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);

 >