def test_diff():
assert Rational(1, 3).diff(x) is S.Zero
assert I.diff(x) is S.Zero
assert pi.diff(x) is S.Zero
assert x.diff(x, 0) == x
assert (x**2).diff(x, 2, x) == 0
assert (x**2).diff((x, 2), x) == 0
assert (x**2).diff((x, 1), x) == 2
assert (x**2).diff((x, 1), (x, 1)) == 2
assert (x**2).diff((x, 2)) == 2
assert (x**2).diff(x, y, 0) == 2*x
assert (x**2).diff(x, (y, 0)) == 2*x
assert (x**2).diff(x, y) == 0
raises(ValueError, lambda: x.diff(1, x))
p = Rational(5)
e = a*b + b**p
assert e.diff(a) == b
assert e.diff(b) == a + 5*b**4
assert e.diff(b).diff(a) == Rational(1)
e = a*(b + c)
assert e.diff(a) == b + c
assert e.diff(b) == a
assert e.diff(b).diff(a) == Rational(1)
e = c**p
assert e.diff(c, 6) == Rational(0)
assert e.diff(c, 5) == Rational(120)
e = c**Rational(2)
assert e.diff(c) == 2*c
e = a*b*c
assert e.diff(c) == a*b
def test_solve_for_functions_derivatives():
t = Symbol('t')
x = Function('x')(t)
y = Function('y')(t)
a11, a12, a21, a22, b1, b2 = symbols('a11,a12,a21,a22,b1,b2')
soln = solve([a11 * x + a12 * y - b1, a21 * x + a22 * y - b2], x, y)
assert soln == {
x: (a22 * b1 - a12 * b2) / (a11 * a22 - a12 * a21),
y: (a11 * b2 - a21 * b1) / (a11 * a22 - a12 * a21),
}
assert solve(x - 1, x) == [1]
assert solve(3 * x - 2, x) == [Rational(2, 3)]
soln = solve([
a11 * x.diff(t) + a12 * y.diff(t) - b1,
a21 * x.diff(t) + a22 * y.diff(t) - b2
], x.diff(t), y.diff(t))
assert soln == {
y.diff(t): (a11 * b2 - a21 * b1) / (a11 * a22 - a12 * a21),
x.diff(t): (a22 * b1 - a12 * b2) / (a11 * a22 - a12 * a21)
}
assert solve(x.diff(t) - 1, x.diff(t)) == [1]
assert solve(3 * x.diff(t) - 2, x.diff(t)) == [Rational(2, 3)]
eqns = set((3 * x - 1, 2 * y - 4))
assert solve(eqns, set((x, y))) == {x: Rational(1, 3), y: 2}
x = Symbol('x')
f = Function('f')
F = x**2 + f(x)**2 - 4 * x - 1
assert solve(F.diff(x), diff(f(x), x)) == [(-x + 2) / f(x)]
# Mixed cased with a Symbol and a Function
x = Symbol('x')
y = Function('y')(t)
soln = solve(
[a11 * x + a12 * y.diff(t) - b1, a21 * x + a22 * y.diff(t) - b2], x,
y.diff(t))
assert soln == {
y.diff(t): (a11 * b2 - a21 * b1) / (a11 * a22 - a12 * a21),
x: (a22 * b1 - a12 * b2) / (a11 * a22 - a12 * a21)
}
def test_solve_for_functions_derivatives():
t = Symbol('t')
x = Function('x')(t)
y = Function('y')(t)
a11, a12, a21, a22, b1, b2 = symbols('a11,a12,a21,a22,b1,b2')
soln = solve([a11*x + a12*y - b1, a21*x + a22*y - b2], x, y)
assert soln == {
x: (a22*b1 - a12*b2)/(a11*a22 - a12*a21),
y: (a11*b2 - a21*b1)/(a11*a22 - a12*a21),
}
assert solve(x - 1, x) == [1]
assert solve(3*x - 2, x) == [Rational(2, 3)]
soln = solve([a11*x.diff(t) + a12*y.diff(t) - b1, a21*x.diff(t) +
a22*y.diff(t) - b2], x.diff(t), y.diff(t))
assert soln == { y.diff(t): (a11*b2 - a21*b1)/(a11*a22 - a12*a21),
x.diff(t): (a22*b1 - a12*b2)/(a11*a22 - a12*a21) }
assert solve(x.diff(t) - 1, x.diff(t)) == [1]
assert solve(3*x.diff(t) - 2, x.diff(t)) == [Rational(2, 3)]
eqns = set((3*x - 1, 2*y - 4))
assert solve(eqns, set((x, y))) == { x: Rational(1, 3), y: 2 }
x = Symbol('x')
f = Function('f')
F = x**2 + f(x)**2 - 4*x - 1
assert solve(F.diff(x), diff(f(x), x)) == [(-x + 2)/f(x)]
# Mixed cased with a Symbol and a Function
x = Symbol('x')
y = Function('y')(t)
soln = solve([a11*x + a12*y.diff(t) - b1, a21*x +
a22*y.diff(t) - b2], x, y.diff(t))
assert soln == { y.diff(t): (a11*b2 - a21*b1)/(a11*a22 - a12*a21),
x: (a22*b1 - a12*b2)/(a11*a22 - a12*a21) }
def test_speed():
# this should return in 0.0s. If it takes forever, it's wrong.
assert x.diff(x, 10**8) == 0
