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MATH02, Derivative calculus

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diff : Derivative of an abstract function

First-order derivative

import sympy
#symypy.init_printing()
from sympy import symbols, Function, diff

x = symbols('x')
f = Function('f')(x)

f.diff(x)              # equivalent to diff(f, x)

OUTPUT : \(\displaystyle \frac{d}{d x} f{\left(x \right)}\)


Higher-order derivative

import sympy
#symypy.init_printing()
from sympy import symbols, Function, diff

x = symbols('x')
f = Function('f')(x)

f.diff(x, 3)           # equivalent to f.diff(x, x, x)
                       # equivalent to diff(f, x, 3)
                       # equivalent to diff(f, x, x, x)

OUTPUT : \(\displaystyle \frac{d^{3}}{d x^{3}} f{\left(x \right)}\)


Derivative of multivariate functions

import sympy
#symypy.init_printing()
from sympy import symbols, Function, diff

x, y = symbols('x, y')
g = sympy.Function('g')(x, y)

g2 = g.diff(x, y)           # equivalent to diff(g, x, y)
g5 = g.diff(x, 3, y, 2)     # equivalent to g.diff(x, x, x, y, y)
                            # equivalent to diff(g, x, 3, y, 2)
                            # equivalent to diff(g, x, x, x, y, y)

g2, g5

OUTPUT : \(\displaystyle \left( \frac{\partial^{2}}{\partial y\partial x} g{\left(x,y \right)}, \ \frac{\partial^{5}}{\partial y^{2}\partial x^{3}} g{\left(x,y \right)}\right)\)


diff : Derivative of an specific function

Polynomials

import sympy
#symypy.init_printing()
from sympy import symbols, diff

x = symbols('x')
expr = x**4 + x**3 + x**2 + x + 1
expr.diff(x)

OUTPUT : \(\displaystyle 4 x^{3} + 3 x^{2} + 2 x + 1\)


Trigonometric

import sympy
#symypy.init_printing()
from sympy import symbols, diff, cos, sin

x, y = symbols('x, y')
expr = sin(x * y) * cos(x / 2)
expr.diff(x)

OUTPUT : \(\displaystyle y \cos{\left(\frac{x}{2} \right)} \cos{\left(x y \right)} - \frac{\sin{\left(\frac{x}{2} \right)} \sin{\left(x y \right)}}{2}\)


Special function

import sympy
#symypy.init_printing()
from sympy import symbols, diff

x = symbols('x')
expr = sympy.special.polynomials.hermite(x, 0)
expr.diff(x)

OUTPUT : \(\displaystyle \frac{2^{x} \sqrt{\pi} \operatorname{polygamma}{\left(0,\frac{1}{2} - \frac{x}{2} \right)}}{2 \Gamma\left(\frac{1}{2} - \frac{x}{2}\right)} + \frac{2^{x} \sqrt{\pi} \log{\left(2 \right)}}{\Gamma\left(\frac{1}{2} - \frac{x}{2}\right)}\)


Derivative

Symbolically represent a derivative

import sympy
#symypy.init_printing()
from sympy import symbols, Derivative, exp, cos

x = symbols('x')
Derivative(exp(cos(x)), x)

OUTPUT : \(\displaystyle \frac{d}{d x} e^{\cos{\left(x \right)}}\)


Evalutation for a derivative

import sympy
#symypy.init_printing()
from sympy import symbols, Derivative, exp, cos

x = symbols('x')
d = Derivative(exp(cos(x)), x)
d.doit()

OUTPUT : \(\displaystyle - e^{\cos{\left(x \right)}} \sin{\left(x \right)}\)


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