In [1]:
from sympy import *
In [8]:
x, y = symbols('x y')
In [9]:
expr = expand((x+y)**6)
In [11]:
expr
Out[11]:
$\displaystyle x^{6} + 6 x^{5} y + 15 x^{4} y^{2} + 20 x^{3} y^{3} + 15 x^{2} y^{4} + 6 x y^{5} + y^{6}$
In [14]:
diff(expr, x)
Out[14]:
$\displaystyle 6 x^{5} + 30 x^{4} y + 60 x^{3} y^{2} + 60 x^{2} y^{3} + 30 x y^{4} + 6 y^{5}$
In [15]:
diff(expr, y)
Out[15]:
$\displaystyle 6 x^{5} + 30 x^{4} y + 60 x^{3} y^{2} + 60 x^{2} y^{3} + 30 x y^{4} + 6 y^{5}$
In [16]:
diff(expr, x, 2)
Out[16]:
$\displaystyle 30 \left(x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4}\right)$
In [17]:
diff(expr, x, x)
Out[17]:
$\displaystyle 30 \left(x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4}\right)$
In [18]:
diff(expr, x, x, y, x)
Out[18]:
$\displaystyle 360 \left(x^{2} + 2 x y + y^{2}\right)$
In [19]:
diff(expr, y, x, x, x)
Out[19]:
$\displaystyle 360 \left(x^{2} + 2 x y + y^{2}\right)$
In [20]:
diff(expr, y, x, 2, y, x)
Out[20]:
$\displaystyle 720 \left(x + y\right)$
In [21]:
diff(sin(x),x)
Out[21]:
$\displaystyle \cos{\left(x \right)}$
In [24]:
diff(sin(x),x, 4)
Out[24]:
$\displaystyle \sin{\left(x \right)}$
In [28]:
integrate(x**2, (x, 0, 1))
Out[28]:
$\displaystyle \frac{1}{3}$
In [29]:
integrate(E ** (-x**2), (x, -oo, oo))
Out[29]:
$\displaystyle \sqrt{\pi}$
In [32]:
integrate(E ** (-x**3), (x, -oo, oo))
Out[32]:
$\displaystyle \int\limits_{-\infty}^{\infty} e^{- x^{3}}\, dx$
In [35]:
4 * integrate(sqrt(1 - x**2), (x, 0, 1))
Out[35]:
$\displaystyle \pi$