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%typeset_mode True


diff(exp(-x^2),x)
−2 xe(−x2)\displaystyle -2 \, x e^{\left(-x^{2}\right)}
'diff(exp(-x^2),x)'
diff(exp(-x^2),x)
diff(exp(-x^2),x,2)
4 x2e(−x2)−2 e(−x2)\displaystyle 4 \, x^{2} e^{\left(-x^{2}\right)} - 2 \, e^{\left(-x^{2}\right)}
diff(log(x/(x^2+1)),x)
−(x2+1)(2 x2(x2+1)2−1x2+1)x\displaystyle -\frac{{\left(x^{2} + 1\right)} {\left(\frac{2 \, x^{2}}{{\left(x^{2} + 1\right)}^{2}} - \frac{1}{x^{2} + 1}\right)}}{x}
diff(log(x/(x^2+1)),x).simplify_full()
−x2−1x3+x\displaystyle -\frac{x^{2} - 1}{x^{3} + x}


diff(x^(x^x),x)
(xx(log⁡(x)+1)log⁡(x)+xxx)x(xx)\displaystyle {\left(x^{x} {\left(\log\left(x\right) + 1\right)} \log\left(x\right) + \frac{x^{x}}{x}\right)} x^{\left(x^{x}\right)}
diff(x^(x^x),x).collect(log(x))
xxx(xx)log⁡(x)2+xxx(xx)log⁡(x)+xxx(xx)x\displaystyle x^{x} x^{\left(x^{x}\right)} \log\left(x\right)^{2} + x^{x} x^{\left(x^{x}\right)} \log\left(x\right) + \frac{x^{x} x^{\left(x^{x}\right)}}{x}
%html
Derivace funkce dané implicitně
Derivace funkce dané implicitně
reset()

var('x')
x\displaystyle x
y = function('y')(x);y
y(x)\displaystyle y\left(x\right)
dy=y.diff(x);dy
∂∂xy(x)\displaystyle \frac{\partial}{\partial x}y\left(x\right)
var('c')
c\displaystyle c
eq=x^2+y^2==c;eq
x2+y(x)2=c\displaystyle x^{2} + y\left(x\right)^{2} = c
dc=diff(eq,x);dc
2 y(x)∂∂xy(x)+2 x=0\displaystyle 2 \, y\left(x\right) \frac{\partial}{\partial x}y\left(x\right) + 2 \, x = 0
yx=solve(dc,dy);yx
[∂∂xy(x)=−xy(x)\displaystyle \frac{\partial}{\partial x}y\left(x\right) = -\frac{x}{y\left(x\right)}]
dcc=diff(eq,x,2);dcc
2 ∂∂xy(x)2+2 y(x)∂2(∂x)2y(x)+2=0\displaystyle 2 \, \frac{\partial}{\partial x}y\left(x\right)^{2} + 2 \, y\left(x\right) \frac{\partial^{2}}{(\partial x)^{2}}y\left(x\right) + 2 = 0
dyy=y.diff(x,2);dyy
∂2(∂x)2y(x)\displaystyle \frac{\partial^{2}}{(\partial x)^{2}}y\left(x\right)
yxx=solve(dcc,dyy);yxx
[∂2(∂x)2y(x)=−∂∂xy(x)2+1y(x)\displaystyle \frac{\partial^{2}}{(\partial x)^{2}}y\left(x\right) = -\frac{\frac{\partial}{\partial x}y\left(x\right)^{2} + 1}{y\left(x\right)}]
yxx[0].rhs().subs(yx).simplify_full()
−x2+y(x)2y(x)3\displaystyle -\frac{x^{2} + y\left(x\right)^{2}}{y\left(x\right)^{3}}
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Parciální derivace
Parciální derivace
var('x,y,a')
(x\displaystyle x, y\displaystyle y, a\displaystyle a)
diff(exp(a*x*y^2),x).diff(y,2)
4 a3x2y4e(axy2)+10 a2xy2e(axy2)+2 ae(axy2)\displaystyle 4 \, a^{3} x^{2} y^{4} e^{\left(a x y^{2}\right)} + 10 \, a^{2} x y^{2} e^{\left(a x y^{2}\right)} + 2 \, a e^{\left(a x y^{2}\right)}
diff(exp(a*x*y^2),x,1,y,2)
4 a3x2y4e(axy2)+10 a2xy2e(axy2)+2 ae(axy2)\displaystyle 4 \, a^{3} x^{2} y^{4} e^{\left(a x y^{2}\right)} + 10 \, a^{2} x y^{2} e^{\left(a x y^{2}\right)} + 2 \, a e^{\left(a x y^{2}\right)}
diff(sin(x+y)/y^4, x,5,y,2)
−cos⁡(x+y)y4+8 sin⁡(x+y)y5+20 cos⁡(x+y)y6\displaystyle -\frac{\cos\left(x + y\right)}{y^{4}} + \frac{8 \, \sin\left(x + y\right)}{y^{5}} + \frac{20 \, \cos\left(x + y\right)}{y^{6}}
var('n')
n\displaystyle n


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Derivace funkce (ve smyslu datové struktury v Sage)
Derivace funkce (ve smyslu datové struktury v Sage)


g(x)=x^n*exp(sin(x));g
x ↦ xnesin⁡(x)\displaystyle x \ {\mapsto}\ x^{n} e^{\sin\left(x\right)}
g.diff()
x ↦ (nxn−1esin⁡(x)+xncos⁡(x)esin⁡(x))\displaystyle x \ {\mapsto}\ \left(n x^{n - 1} e^{\sin\left(x\right)} + x^{n} \cos\left(x\right) e^{\sin\left(x\right)}\right)
g.diff()(pi/6)
((16 π)n−1ne12+12 3(16 π)ne12)\displaystyle \left(\left(\frac{1}{6} \, \pi\right)^{n - 1} n e^{\frac{1}{2}} + \frac{1}{2} \, \sqrt{3} \left(\frac{1}{6} \, \pi\right)^{n} e^{\frac{1}{2}}\right)
g.diff().diff()
(x ↦ (n−1)nxn−2esin⁡(x)+2 nxn−1cos⁡(x)esin⁡(x)+xncos⁡(x)2esin⁡(x)−xnesin⁡(x)sin⁡(x))\displaystyle \left(\begin{array}{r} x \ {\mapsto}\ {\left(n - 1\right)} n x^{n - 2} e^{\sin\left(x\right)} + 2 \, n x^{n - 1} \cos\left(x\right) e^{\sin\left(x\right)} + x^{n} \cos\left(x\right)^{2} e^{\sin\left(x\right)} - x^{n} e^{\sin\left(x\right)} \sin\left(x\right) \end{array}\right)
f(x,y)=x^2*y+2*x*y+x

f.diff() #vypočte obě parciální derivace
(x,y) ↦ (2 xy+2 y+1, x2+2 x)\displaystyle \left( x, y \right) \ {\mapsto} \ \left(2 \, x y + 2 \, y + 1,\,x^{2} + 2 \, x\right)
f.diff(x); f.diff(y)
(x,y) ↦ 2 xy+2 y+1\displaystyle \left( x, y \right) \ {\mapsto} \ 2 \, x y + 2 \, y + 1
(x,y) ↦ x2+2 x\displaystyle \left( x, y \right) \ {\mapsto} \ x^{2} + 2 \, x
f.diff(x).diff(y) #smíšená parciální derivace vzhledem k x a y.
(x,y) ↦ 2 x+2\displaystyle \left( x, y \right) \ {\mapsto} \ 2 \, x + 2
F=piecewise([[(-infinity,0),sin(x)],[[0,0],arctan(x)],[(0,infinity),arctan(x)]], var=x)

plot(F(x), (x,-3*pi,3*pi))

F=piecewise([[(-infinity,0),sin(x)],[[0,0],arctan(x)],[(0,infinity),arctan(x)]], var=x)

F.derivative()
(0, 0, 0)*D[0]piecewise(x|-->sin(x) on (-oo, 0), x|-->arctan(x) on {0}, x|-->arctan(x) on (0, +oo); x) + D[1]piecewise(x|-->sin(x) on (-oo, 0), x|-->arctan(x) on {0}, x|-->arctan(x) on (0, +oo); x) 
integral(x/(x^3+1), x, hold=True)
∫xx3+1 dx\displaystyle \int \frac{x}{x^{3} + 1}\,{d x}


i=integral(x/(x^3+1), x);i
13 3arctan⁡(13 3(2 x−1))+16 log⁡(x2−x+1)−13 log⁡(x+1)\displaystyle \frac{1}{3} \, \sqrt{3} \arctan\left(\frac{1}{3} \, \sqrt{3} {\left(2 \, x - 1\right)}\right) + \frac{1}{6} \, \log\left(x^{2} - x + 1\right) - \frac{1}{3} \, \log\left(x + 1\right)
diff(i,x).simplify_full()
xx3+1\displaystyle \frac{x}{x^{3} + 1}
show(integrate(x/(x^3+1), x, hold=True), '=', i)
∫xx3+1 dx\displaystyle \int \frac{x}{x^{3} + 1}\,{d x} = 13 3arctan⁡(13 3(2 x−1))+16 log⁡(x2−x+1)−13 log⁡(x+1)\displaystyle \frac{1}{3} \, \sqrt{3} \arctan\left(\frac{1}{3} \, \sqrt{3} {\left(2 \, x - 1\right)}\right) + \frac{1}{6} \, \log\left(x^{2} - x + 1\right) - \frac{1}{3} \, \log\left(x + 1\right)
i=integral(x/(x^5+1),x);i
−2 5arctan⁡(4 x+5−12 5+10)5 2 5+10+2 5arctan⁡(4 x−5−1−2 5+10)5 −2 5+10−log⁡(2 x2−x(5+1)+2)5 (5+1)+log⁡(2 x2+x(5−1)+2)5 (5−1)−15 log⁡(x+1)\displaystyle -\frac{2 \, \sqrt{5} \arctan\left(\frac{4 \, x + \sqrt{5} - 1}{\sqrt{2 \, \sqrt{5} + 10}}\right)}{5 \, \sqrt{2 \, \sqrt{5} + 10}} + \frac{2 \, \sqrt{5} \arctan\left(\frac{4 \, x - \sqrt{5} - 1}{\sqrt{-2 \, \sqrt{5} + 10}}\right)}{5 \, \sqrt{-2 \, \sqrt{5} + 10}} - \frac{\log\left(2 \, x^{2} - x {\left(\sqrt{5} + 1\right)} + 2\right)}{5 \, {\left(\sqrt{5} + 1\right)}} + \frac{\log\left(2 \, x^{2} + x {\left(\sqrt{5} - 1\right)} + 2\right)}{5 \, {\left(\sqrt{5} - 1\right)}} - \frac{1}{5} \, \log\left(x + 1\right)
diff(i,x).simplify_full()
xx5+1\displaystyle \frac{x}{x^{5} + 1}
integral(2*x*(x^2+1)^24, x)
125 (x2+1)25\displaystyle \frac{1}{25} \, {\left(x^{2} + 1\right)}^{25}
Určitý integrál
i=integral(x/(x^3+1), (x,1,2));i
118 3π−16 log⁡(3)+13 log⁡(2)\displaystyle \frac{1}{18} \, \sqrt{3} \pi - \frac{1}{6} \, \log\left(3\right) + \frac{1}{3} \, \log\left(2\right)
i.n()
0.350246906114333\displaystyle 0.350246906114333
numerical_integral(x/(x^3+1), 1,2)
(0.3502469061143331\displaystyle 0.3502469061143331, 3.888521794718582×10−15\displaystyle 3.888521794718582 \times 10^{-15})
integral(1/x^2, (x,-1,1))
Error in lines 1-1 Traceback (most recent call last): File "/ext/sage/10.3/src/sage/interfaces/maxima_lib.py", line 817, in sr_integral return max_to_sr(maxima_eval(([max_integrate], ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ File "sage/libs/ecl.pyx", line 838, in sage.libs.ecl.EclObject.__call__ return ecl_wrap(ecl_safe_apply(self.obj, (<EclObject>lispargs).obj)) File "sage/libs/ecl.pyx", line 358, in sage.libs.ecl.ecl_safe_apply raise RuntimeError("ECL says: {}".format(message)) RuntimeError: ECL says: defint: integral is divergent. During handling of the above exception, another exception occurred: Traceback (most recent call last): File "/cocalc/lib/python3.11/site-packages/smc_sagews/sage_server.py", line 1244, in execute exec( File "", line 1, in <module> File "/ext/sage/10.3/src/sage/misc/functional.py", line 788, in integral return x.integral(*args, **kwds) ^^^^^^^^^^^^^^^^^^^^^^^^^ File "sage/symbolic/expression.pyx", line 13268, in sage.symbolic.expression.Expression.integral return integral(self, *args, **kwds) File "/ext/sage/10.3/src/sage/symbolic/integration/integral.py", line 1066, in integrate return definite_integral(expression, v, a, b, hold=hold) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ File "sage/symbolic/function.pyx", line 1047, in sage.symbolic.function.BuiltinFunction.__call__ res = super().__call__( File "sage/symbolic/function.pyx", line 555, in sage.symbolic.function.Function.__call__ return call_registered_function(self._serial, self._nargs, args, hold, File "sage/symbolic/pynac_function_impl.pxi", line 1, in sage.symbolic.expression.call_registered_function cpdef call_registered_function(unsigned serial, File "sage/symbolic/pynac_function_impl.pxi", line 47, in sage.symbolic.expression.call_registered_function res = g_function_evalv(serial, vec, hold) File "sage/symbolic/function.pyx", line 1137, in sage.symbolic.function.BuiltinFunction._evalf_or_eval_ return self._eval0_(*args) File "/ext/sage/10.3/src/sage/symbolic/integration/integral.py", line 254, in _eval_ A = integrator(*args) ^^^^^^^^^^^^^^^^^ File "/ext/sage/10.3/src/sage/symbolic/integration/external.py", line 46, in maxima_integrator result = maxima.sr_integral(expression, v, a, b) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ File "/ext/sage/10.3/src/sage/interfaces/maxima_lib.py", line 825, in sr_integral raise ValueError("Integral is divergent.") ValueError: Integral is divergent. 
plot(1/x^2, (x,-1,1), ymin=0, ymax=10)

%html
Nevlastní integrál
Nevlastní integrál
var('t')
t\displaystyle t
integral(t^4*log(t)^2/(1+3*t^2)^3, t,0,oo)
1576 3π3+1576 3πlog⁡(3)2−1108 3πlog⁡(3)+1216 3π\displaystyle \frac{1}{576} \, \sqrt{3} \pi^{3} + \frac{1}{576} \, \sqrt{3} \pi \log\left(3\right)^{2} - \frac{1}{108} \, \sqrt{3} \pi \log\left(3\right) + \frac{1}{216} \, \sqrt{3} \pi
integral(exp(arcsin(x)), x,0,1)
12 e(12 π)−12\displaystyle \frac{1}{2} \, e^{\left(\frac{1}{2} \, \pi\right)} - \frac{1}{2}
integral(sqrt(9-x^2), x)
12 −x2+9x+92 arcsin⁡(13 x)\displaystyle \frac{1}{2} \, \sqrt{-x^{2} + 9} x + \frac{9}{2} \, \arcsin\left(\frac{1}{3} \, x\right)
Sumace
var('k,n,a,y')
(k\displaystyle k, n\displaystyle n, a\displaystyle a, y\displaystyle y)
sum(k^7, k,1,20, hold=True)
∑k=120k7\displaystyle {\sum_{k=1}^{20} k^{7}}
show(sum(k^7, k,1,20, hold=True), '=', sum(k^7, k,1,20))
∑k=120k7\displaystyle {\sum_{k=1}^{20} k^{7}} = 3877286700\displaystyle 3877286700
sum(k^7, k,1,n)
18 n8+12 n7+712 n6−724 n4+112 n2\displaystyle \frac{1}{8} \, n^{8} + \frac{1}{2} \, n^{7} + \frac{7}{12} \, n^{6} - \frac{7}{24} \, n^{4} + \frac{1}{12} \, n^{2}
sum(1/(k^2-4), k,3,oo)
2548\displaystyle \frac{25}{48}
Taylorova řada


t=taylor(sin(tan(x))-tan(sin(x)), x,0,25);t
480743327105054114616431560515174400000 x25−3746946250748836690480522485760000 x23−20975554600017602818775552000 x21−16641083631905468364800 x19−101932074358914560 x17−31114886954486432000 x15−957392 x13−191375600 x11−29756 x9−130 x7\displaystyle \frac{48074332710505411}{4616431560515174400000} \, x^{25} - \frac{374694625074883}{6690480522485760000} \, x^{23} - \frac{2097555460001}{7602818775552000} \, x^{21} - \frac{1664108363}{1905468364800} \, x^{19} - \frac{10193207}{4358914560} \, x^{17} - \frac{311148869}{54486432000} \, x^{15} - \frac{95}{7392} \, x^{13} - \frac{1913}{75600} \, x^{11} - \frac{29}{756} \, x^{9} - \frac{1}{30} \, x^{7}
type(t)
<class 'sage.symbolic.expression.Expression'> 
f=sin(tan(x))-tan(sin(x))

t1=f.series(x,25);t1
(−130)x7+(−29756)x9+(−191375600)x11+(−957392)x13+(−31114886954486432000)x15+(−101932074358914560)x17+(−16641083631905468364800)x19+(−20975554600017602818775552000)x21+(−3746946250748836690480522485760000)x23+O(x25)\displaystyle {(-\frac{1}{30})} x^{7} + {(-\frac{29}{756})} x^{9} + {(-\frac{1913}{75600})} x^{11} + {(-\frac{95}{7392})} x^{13} + {(-\frac{311148869}{54486432000})} x^{15} + {(-\frac{10193207}{4358914560})} x^{17} + {(-\frac{1664108363}{1905468364800})} x^{19} + {(-\frac{2097555460001}{7602818775552000})} x^{21} + {(-\frac{374694625074883}{6690480522485760000})} x^{23} + \mathcal{O}\left(x^{25}\right)
type(t1)
<class 'sage.symbolic.expression.SymbolicSeries'> 


t1.coefficients()
[[0\displaystyle 0, 0\displaystyle 0], [0\displaystyle 0, 1\displaystyle 1], [0\displaystyle 0, 2\displaystyle 2], [0\displaystyle 0, 3\displaystyle 3], [0\displaystyle 0, 4\displaystyle 4], [0\displaystyle 0, 5\displaystyle 5], [0\displaystyle 0, 6\displaystyle 6], [−130\displaystyle -\frac{1}{30}, 7\displaystyle 7], [0\displaystyle 0, 8\displaystyle 8], [−29756\displaystyle -\frac{29}{756}, 9\displaystyle 9], [0\displaystyle 0, 10\displaystyle 10], [−191375600\displaystyle -\frac{1913}{75600}, 11\displaystyle 11], [0\displaystyle 0, 12\displaystyle 12], [−957392\displaystyle -\frac{95}{7392}, 13\displaystyle 13], [0\displaystyle 0, 14\displaystyle 14], [−31114886954486432000\displaystyle -\frac{311148869}{54486432000}, 15\displaystyle 15], [0\displaystyle 0, 16\displaystyle 16], [−101932074358914560\displaystyle -\frac{10193207}{4358914560}, 17\displaystyle 17], [0\displaystyle 0, 18\displaystyle 18], [−16641083631905468364800\displaystyle -\frac{1664108363}{1905468364800}, 19\displaystyle 19], [0\displaystyle 0, 20\displaystyle 20], [−20975554600017602818775552000\displaystyle -\frac{2097555460001}{7602818775552000}, 21\displaystyle 21], [0\displaystyle 0, 22\displaystyle 22], [−3746946250748836690480522485760000\displaystyle -\frac{374694625074883}{6690480522485760000}, 23\displaystyle 23], [0\displaystyle 0, 24\displaystyle 24]]
t2=t1.truncate();t2
−3746946250748836690480522485760000 x23−20975554600017602818775552000 x21−16641083631905468364800 x19−101932074358914560 x17−31114886954486432000 x15−957392 x13−191375600 x11−29756 x9−130 x7\displaystyle -\frac{374694625074883}{6690480522485760000} \, x^{23} - \frac{2097555460001}{7602818775552000} \, x^{21} - \frac{1664108363}{1905468364800} \, x^{19} - \frac{10193207}{4358914560} \, x^{17} - \frac{311148869}{54486432000} \, x^{15} - \frac{95}{7392} \, x^{13} - \frac{1913}{75600} \, x^{11} - \frac{29}{756} \, x^{9} - \frac{1}{30} \, x^{7}
type(t2)
<class 'sage.symbolic.expression.Expression'> 
taylor(x*y^3,(x,1),(y,-1),4)
(x−1)(y+1)3−3 (x−1)(y+1)2+(y+1)3+3 (x−1)(y+1)−3 (y+1)2−x+3 y+3\displaystyle {\left(x - 1\right)} {\left(y + 1\right)}^{3} - 3 \, {\left(x - 1\right)} {\left(y + 1\right)}^{2} + {\left(y + 1\right)}^{3} + 3 \, {\left(x - 1\right)} {\left(y + 1\right)} - 3 \, {\left(y + 1\right)}^{2} - x + 3 \, y + 3
%html
Výpočty limit
Výpočty limit
limit( cos(x)^(1/x^3), x=0)
und\displaystyle \mathit{und}
limit( cos(x)^(1/x^3), x=0, dir='plus')
0\displaystyle 0
limit( cos(x)^(1/x^3), x=0, dir='minus')
+∞\displaystyle +\infty
var('a,b')
(a\displaystyle a, b\displaystyle b)
y=exp(a*x)*cos(b*x);y
cos⁡(bx)e(ax)\displaystyle \cos\left(b x\right) e^{\left(a x\right)}
assume(a>0, b>0)



limit(y, x=-oo)
0\displaystyle 0
%html
Diferenciální rovnice
Diferenciální rovnice
y = function('y')(x)

ode=x*diff(y,x)==y;ode
x∂∂xy(x)=y(x)\displaystyle x \frac{\partial}{\partial x}y\left(x\right) = y\left(x\right)
sol=desolve(ode,y);expand(sol)
Cx\displaystyle C x
desolve(ode, y, ics=[1,1])
x\displaystyle x