def test_issue_7231():
from sympy.abc import a
ans1 = f(x).series(x, a)
res = (f(a) + (-a + x) * Subs(Derivative(f(y), y), y, a) +
(-a + x)**2 * Subs(Derivative(f(y), y, y), y, a) / 2 +
(-a + x)**3 * Subs(Derivative(f(y), y, y, y), y, a) / 6 +
(-a + x)**4 * Subs(Derivative(f(y), y, y, y, y), y, a) / 24 +
(-a + x)**5 * Subs(Derivative(f(y), y, y, y, y, y), y, a) / 120 + O(
(-a + x)**6, (x, a)))
assert res == ans1
ans2 = f(x).series(x, a)
assert res == ans2
def test_issue_7068():
from sympy.abc import a, b, f
y1 = Dummy('y')
y2 = Dummy('y')
func1 = f(a + y1 * b)
func2 = f(a + y2 * b)
func1_y = func1.diff(y1)
func2_y = func2.diff(y2)
assert func1_y != func2_y
z1 = Subs(f(a), a, y1)
z2 = Subs(f(a), a, y2)
assert z1 != z2
def test_deriv1():
# These all requre derivatives evaluated at a point (issue 1620) to work.
# See issue 1525
assert f(2*x).diff(x) == 2*Subs(Derivative(f(x), x), Tuple(x), Tuple(2*x))
assert (f(x)**3).diff(x) == 3*f(x)**2*f(x).diff(x)
assert (f(2*x)**3).diff(x) == 6*f(2*x)**2*Subs(Derivative(f(x), x), Tuple(x),
Tuple(2*x))
assert f(2+x).diff(x) == Subs(Derivative(f(x), x), Tuple(x), Tuple(x + 2))
assert f(2+3*x).diff(x) == 3*Subs(Derivative(f(x), x), Tuple(x),
Tuple(3*x + 2))
assert f(3*sin(x)).diff(x) == 3*cos(x)*Subs(Derivative(f(x), x),
Tuple(x), Tuple(3*sin(x)))
def test_Subs_subs():
assert Subs(x * y, x, x).subs(x, y) == Subs(x * y, x, y)
assert Subs(x*y, x, x + 1).subs(x, y) == \
Subs(x*y, x, y + 1)
assert Subs(x*y, y, x + 1).subs(x, y) == \
Subs(y**2, y, y + 1)
a = Subs(x * y * z, (y, x, z), (x + 1, x + z, x))
b = Subs(x * y * z, (y, x, z), (x + 1, y + z, y))
assert a.subs(x, y) == b and \
a.doit().subs(x, y) == a.subs(x, y).doit()
f = Function('f')
g = Function('g')
assert Subs(2 * f(x, y) + g(x), f(x, y),
1).subs(y, 2) == Subs(2 * f(x, y) + g(x), (f(x, y), y), (1, 2))
def test_issue_7231():
from sympy.abc import a
ans1 = f(x).series(x, a)
_xi_1 = ans1.atoms(Dummy).pop()
res = (f(a) + (-a + x)*Subs(Derivative(f(_xi_1), _xi_1), (_xi_1,), (a,)) +
(-a + x)**2*Subs(Derivative(f(_xi_1), _xi_1, _xi_1), (_xi_1,), (a,))/2 +
(-a + x)**3*Subs(Derivative(f(_xi_1), _xi_1, _xi_1, _xi_1),
(_xi_1,), (a,))/6 +
(-a + x)**4*Subs(Derivative(f(_xi_1), _xi_1, _xi_1, _xi_1, _xi_1),
(_xi_1,), (a,))/24 +
(-a + x)**5*Subs(Derivative(f(_xi_1), _xi_1, _xi_1, _xi_1, _xi_1, _xi_1),
(_xi_1,), (a,))/120 + O((-a + x)**6, (x, a)))
assert res == ans1
ans2 = f(x).series(x, a)
assert res == ans2
def check_solutions(eq):
"""
Determines whether solutions returned by diophantine() satisfy the original
equation. Hope to generalize this so we can remove functions like check_ternay_quadratic,
check_solutions_normal, check_solutions()
"""
s = diophantine(eq)
terms = factor_list(eq)[1]
var = list(eq.free_symbols)
var.sort(key=default_sort_key)
okay = True
while len(s) and okay:
solution = s.pop()
okay = False
for term in terms:
subeq = term[0]
if simplify(_mexpand(Subs(subeq, var, solution).doit())) == 0:
okay = True
break
return okay
def is_pell_transformation_ok(eq):
"""
Test whether X*Y, X, or Y terms are present in the equation
after transforming the equation using the transformation returned
by transformation_to_pell(). If they are not present we are good.
Moreover, coefficient of X**2 should be a divisor of coefficient of
Y**2 and the constant term.
"""
A, B = transformation_to_DN(eq)
u = (A * Matrix([X, Y]) + B)[0]
v = (A * Matrix([X, Y]) + B)[1]
simplified = _mexpand(Subs(eq, (x, y), (u, v)).doit())
coeff = dict(
[reversed(t.as_independent(*[X, Y])) for t in simplified.args])
for term in [X * Y, X, Y]:
if term in coeff.keys():
return False
for term in [X**2, Y**2, Integer(1)]:
if term not in coeff.keys():
coeff[term] = Integer(0)
if coeff[X**2] != 0:
return isinstance(S(coeff[Y**2]) / coeff[X**2],
Integer) and isinstance(
S(coeff[Integer(1)]) / coeff[X**2], Integer)
return True
def get_gamma(k,alpha):
f49 = N(Subs(f49Coeff, [_j,_alpha] , [k-1,alpha]))
j = int(np.ceil(k/2))
f27,f31 = get_fCoeffs(j,alpha)
# Eq. 58, Hadden & Lithwick '16
gamma = f49 * (f27*f27 + f31*f31) / f27 / f31 / 2
return gamma
def test_issue_15084_13166():
eq = f(x, g(x))
assert eq.diff((g(x), y)) == Derivative(f(x, g(x)), (g(x), y))
# issue 13166
assert eq.diff(x, 2).doit() == (
(Derivative(f(x, g(x)), (g(x), 2))*Derivative(g(x), x) +
Subs(Derivative(f(x, _xi_2), _xi_2, x), _xi_2, g(x)))*Derivative(g(x),
x) + Derivative(f(x, g(x)), g(x))*Derivative(g(x), (x, 2)) +
Derivative(g(x), x)*Subs(Derivative(f(_xi_1, g(x)), _xi_1, g(x)),
_xi_1, x) + Subs(Derivative(f(_xi_1, g(x)), (_xi_1, 2)), _xi_1, x))
# issue 6681
assert diff(f(x, t, g(x, t)), x).doit() == (
Derivative(f(x, t, g(x, t)), g(x, t))*Derivative(g(x, t), x) +
Subs(Derivative(f(_xi_1, t, g(x, t)), _xi_1), _xi_1, x))
# make sure the order doesn't matter when using diff
assert eq.diff(x, g(x)) == eq.diff(g(x), x)
def test_diff_wrt():
fx = f(x)
dfx = diff(f(x), x)
ddfx = diff(f(x), x, x)
assert diff(sin(fx) + fx**2, fx) == cos(fx) + 2 * fx
assert diff(sin(dfx) + dfx**2, dfx) == cos(dfx) + 2 * dfx
assert diff(sin(ddfx) + ddfx**2, ddfx) == cos(ddfx) + 2 * ddfx
assert diff(fx**2, dfx) == 0
assert diff(fx**2, ddfx) == 0
assert diff(dfx**2, fx) == 0
assert diff(dfx**2, ddfx) == 0
assert diff(ddfx**2, dfx) == 0
assert diff(fx * dfx * ddfx, fx) == dfx * ddfx
assert diff(fx * dfx * ddfx, dfx) == fx * ddfx
assert diff(fx * dfx * ddfx, ddfx) == fx * dfx
assert diff(f(x), x).diff(f(x)) == 0
assert (sin(f(x)) - cos(diff(f(x), x))).diff(f(x)) == cos(f(x))
assert diff(sin(fx), fx, x) == diff(sin(fx), x, fx)
# Chain rule cases
assert f(g(x)).diff(x) == \
Derivative(g(x), x)*Derivative(f(g(x)), g(x))
assert diff(f(g(x), h(y)), x) == \
Derivative(g(x), x)*Derivative(f(g(x), h(y)), g(x))
assert diff(f(g(x), h(x)),
x) == (Derivative(f(g(x), h(x)), g(x)) * Derivative(g(x), x) +
Derivative(f(g(x), h(x)), h(x)) * Derivative(h(x), x))
assert f(sin(x)).diff(x) == cos(x) * Subs(Derivative(f(x), x), x, sin(x))
assert diff(f(g(x)), g(x)) == Derivative(f(g(x)), g(x))
def test_series_of_Subs():
from sympy.abc import x, y, z
subs1 = Subs(sin(x), x, y)
subs2 = Subs(sin(x) * cos(z), x, y)
subs3 = Subs(sin(x * z), (x, z), (y, x))
assert subs1.series(x) == subs1
assert subs1.series(y) == Subs(x, x, y) + Subs(-x**3 / 6, x, y) + Subs(
x**5 / 120, x, y) + O(y**6)
assert subs1.series(z) == subs1
assert subs2.series(z) == Subs(z**4 * sin(x) / 24, x, y) + Subs(
-z**2 * sin(x) / 2, x, y) + Subs(sin(x), x, y) + O(z**6)
assert subs3.series(x).doit() == subs3.doit().series(x)
assert subs3.series(z).doit() == sin(x * y)
def is_solution_quad(var, coeff, u, v):
"""
Check whether (u, v) is solution to the quadratic diophantine equation.
"""
x = var[0]
y = var[1]
eq = x**2*coeff[x**2] + x*y*coeff[x*y] + y**2*coeff[y**2] + x*coeff[x] + y*coeff[y] + coeff[Integer(1)]
return simplify(Subs(eq, (x, y), (u, v)).doit()) == 0
def test_simultaneous_subs():
reps = {x: 0, y: 0}
assert (x / y).subs(reps) != (y / x).subs(reps)
assert (x/y).subs(reps, simultaneous=True) == \
(y/x).subs(reps, simultaneous=True)
reps = reps.items()
assert (x / y).subs(reps) != (y / x).subs(reps)
assert (x/y).subs(reps, simultaneous=True) == \
(y/x).subs(reps, simultaneous=True)
assert Derivative(x, y, z).subs(reps, simultaneous=True) == \
Subs(Derivative(0, y, z), y, 0)
def is_normal_transformation_ok(eq):
A = transformation_to_normal(eq)
X, Y, Z = A * Matrix([x, y, z])
simplified = _mexpand(Subs(eq, (x, y, z), (X, Y, Z)).doit())
coeff = dict(
[reversed(t.as_independent(*[X, Y, Z])) for t in simplified.args])
for term in [X * Y, Y * Z, X * Z]:
if term in coeff.keys():
return False
return True
def test_issue_12005():
e1 = Subs(Derivative(f(x), x), (x,), (x,))
assert e1.diff(x) == Derivative(f(x), x, x)
e2 = Subs(Derivative(f(x), x), (x,), (x**2 + 1,))
assert e2.diff(x) == 2*x*Subs(Derivative(f(x), x, x), (x,), (x**2 + 1,))
e3 = Subs(Derivative(f(x) + y**2 - y, y), (y,), (y**2,))
assert e3.diff(y) == 4*y
e4 = Subs(Derivative(f(x + y), y), (y,), (x**2))
assert e4.diff(y) == S.Zero
e5 = Subs(Derivative(f(x), x), (y, z), (y, z))
assert e5.diff(x) == Derivative(f(x), x, x)
assert f(g(x)).diff(g(x), g(x)) == Subs(Derivative(f(y), y, y), (y,), (g(x),))
def test_subs_in_derivative():
expr = sin(x*exp(y))
u = Function('u')
v = Function('v')
assert Derivative(expr, y).subs(y, x).doit() == \
Derivative(expr, y).doit().subs(y, x)
assert Derivative(f(x, y), y).subs(y, x) == Subs(Derivative(f(x, y), y), y, x)
assert Derivative(f(x, y), y).subs(x, y) == Subs(Derivative(f(x, y), y), x, y)
assert Derivative(f(x, y), y).subs(y, g(x, y)) == Subs(Derivative(f(x, y), y), y, g(x, y))
assert Derivative(f(x, y), y).subs(x, g(x, y)) == Subs(Derivative(f(x, y), y), x, g(x, y))
assert Derivative(f(u(x), h(y)), h(y)).subs(h(y), g(x, y)) == \
Subs(Derivative(f(u(x), h(y)), h(y)), h(y), g(x, y))
assert Derivative(f(x, y), y).subs(y, z) == Derivative(f(x, z), z)
assert Derivative(f(x, y), y).subs(y, g(y)) == Derivative(f(x, g(y)), g(y))
assert Derivative(f(g(x), h(y)), h(y)).subs(h(y), u(y)) == \
Derivative(f(g(x), u(y)), u(y))
# Issue 13791. No comparison (it's a long formula) but this used to raise an exception.
assert isinstance(v(x, y, u(x, y)).diff(y).diff(x).diff(y), Expr)
# This is also related to issues 13791 and 13795
F = Lambda((x, y), exp(2*x + 3*y))
abstract = f(x, f(x, x)).diff(x, 2)
concrete = F(x, F(x, x)).diff(x, 2)
assert (abstract.replace(f, F).doit() - concrete).simplify() == 0
def test_complicated_derivative_with_Indexed():
x, y = symbols("x,y", cls=IndexedBase)
sigma = symbols("sigma")
i, j, k = symbols("i,j,k")
m0, m1, m2, m3, m4, m5 = symbols("m0:6")
f = Function("f")
expr = f((x[i] - y[i]) ** 2 / sigma)
_xi_1 = symbols("xi_1", cls=Dummy)
assert expr.diff(x[m0]).dummy_eq(
(x[i] - y[i])
* KroneckerDelta(i, m0)
* 2
* Subs(Derivative(f(_xi_1), _xi_1), (_xi_1,), ((x[i] - y[i]) ** 2 / sigma,))
/ sigma
)
assert (
expr.diff(x[m0])
.diff(x[m1])
.dummy_eq(
2
* KroneckerDelta(i, m0)
* KroneckerDelta(i, m1)
* Subs(Derivative(f(_xi_1), _xi_1), (_xi_1,), ((x[i] - y[i]) ** 2 / sigma,))
/ sigma
+ 4
* (x[i] - y[i]) ** 2
* KroneckerDelta(i, m0)
* KroneckerDelta(i, m1)
* Subs(
Derivative(f(_xi_1), _xi_1, _xi_1),
(_xi_1,),
((x[i] - y[i]) ** 2 / sigma,),
)
/ sigma ** 2
)
)
def test_deriv1():
# These all requre derivatives evaluated at a point (issue 4719) to work.
# See issue 4624
assert f(2*x).diff(x) == 2*Subs(Derivative(f(x), x), Tuple(x), Tuple(2*x))
assert (f(x)**3).diff(x) == 3*f(x)**2*f(x).diff(x)
assert (
f(2*x)**3).diff(x) == 6*f(2*x)**2*Subs(Derivative(f(x), x), Tuple(x),
Tuple(2*x))
assert f(2 + x).diff(x) == Subs(Derivative(f(x), x), Tuple(x), Tuple(x + 2))
assert f(2 + 3*x).diff(x) == 3*Subs(Derivative(f(x), x), Tuple(x),
Tuple(3*x + 2))
assert f(3*sin(x)).diff(x) == 3*cos(x)*Subs(Derivative(f(x), x),
Tuple(x), Tuple(3*sin(x)))
# See issue 8510
assert f(x, x + z).diff(x) == Subs(Derivative(f(y, x + z), y), Tuple(y), Tuple(x)) \
+ Subs(Derivative(f(x, y), y), Tuple(y), Tuple(x + z))
assert f(x, x**2).diff(x) == Subs(Derivative(f(y, x**2), y), Tuple(y), Tuple(x)) \
+ 2*x*Subs(Derivative(f(x, y), y), Tuple(y), Tuple(x**2))
def solutions_ok_quadratic(eq):
"""
Determines whether solutions returned by diop_solve() satisfy the original
equation.
"""
s = diop_solve(eq)
x, y = symbols("x, y", Integer=True)
ok = True
while len(s) and ok:
u, v = s.pop()
if simplify(simplify(Subs(eq, (x, y), (u, v)).doit())) != 0:
ok = False
return ok
def test_derivative_subs():
f = Function('f')
g = Function('g')
assert Derivative(f(x), x).subs(f(x), y) != 0
# need xreplace to put the function back, see #13803
assert Derivative(f(x), x).subs(f(x), y).xreplace({y: f(x)}) == \
Derivative(f(x), x)
# issues 5085, 5037
assert cse(Derivative(f(x), x) + f(x))[1][0].has(Derivative)
assert cse(Derivative(f(x, y), x) +
Derivative(f(x, y), y))[1][0].has(Derivative)
eq = Derivative(g(x), g(x))
assert eq.subs(g, f) == Derivative(f(x), f(x))
assert eq.subs(g(x), f(x)) == Derivative(f(x), f(x))
assert eq.subs(g, cos) == Subs(Derivative(y, y), y, cos(x))
def test_dont_cse_tuples():
from sympy import Subs
f = Function("f")
g = Function("g")
name_val, (expr, ) = cse(
Subs(f(x, y), (x, y), (0, 1)) + Subs(g(x, y), (x, y), (0, 1)))
assert name_val == []
assert expr == (Subs(f(x, y), (x, y),
(0, 1)) + Subs(g(x, y), (x, y), (0, 1)))
name_val, (expr, ) = cse(
Subs(f(x, y), (x, y), (0, x + y)) + Subs(g(x, y), (x, y), (0, x + y)))
assert name_val == [(x0, x + y)]
assert expr == Subs(f(x, y), (x, y), (0, x0)) + \
Subs(g(x, y), (x, y), (0, x0))
def test_deriv1():
# These all require derivatives evaluated at a point (issue 4719) to work.
# See issue 4624
assert f(2 * x).diff(x) == 2 * Subs(Derivative(f(x), x), x, 2 * x)
assert (f(x)**3).diff(x) == 3 * f(x)**2 * f(x).diff(x)
assert (f(
2 *
x)**3).diff(x) == 6 * f(2 * x)**2 * Subs(Derivative(f(x), x), x, 2 * x)
assert f(2 + x).diff(x) == Subs(Derivative(f(x), x), x, x + 2)
assert f(2 + 3 * x).diff(x) == 3 * Subs(Derivative(f(x), x), x, 3 * x + 2)
assert f(3 * sin(x)).diff(x) == 3 * cos(x) * Subs(Derivative(f(x), x), x,
3 * sin(x))
# See issue 8510
assert f(x, x + z).diff(x) == (Subs(Derivative(f(y, x + z), y), y, x) +
Subs(Derivative(f(x, y), y), y, x + z))
assert f(x,
x**2).diff(x) == (2 * x * Subs(Derivative(f(x, y), y), y, x**2) +
Subs(Derivative(f(y, x**2), y), y, x))
# but Subs is not always necessary
assert f(x, g(y)).diff(g(y)) == Derivative(f(x, g(y)), g(y))
def test_879():
f = Function('f')
assert f(x).series(x, 0, 3, dir='-') == \
f(0) + x*Subs(Derivative(f(x), x), (x,), (0,)) + \
x**2*Subs(Derivative(f(x), x, x), (x,), (0,))/2 + O(x**3)
assert f(x).series(x, 0, 3) == \
f(0) + x*Subs(Derivative(f(x), x), (x,), (0,)) + \
x**2*Subs(Derivative(f(x), x, x), (x,), (0,))/2 + O(x**3)
assert f(x**2).series(x, 0, 3) == \
f(0) + x**2*Subs(Derivative(f(x), x), (x,), (0,)) + O(x**3)
assert f(x**2+1).series(x, 0, 3) == \
f(1) + x**2*Subs(Derivative(f(x), x), (x,), (1,)) + O(x**3)
class TestF(Function):
pass
assert TestF(x).series(x, 0, 3) == TestF(0) + \
x*Subs(Derivative(TestF(x), x), (x,), (0,)) + \
x**2*Subs(Derivative(TestF(x), x, x), (x,), (0,))/2 + O(x**3)
def test_issue_3978():
f = Function("f")
assert f(x).series(x, 0, 3, dir="-") == f(0) + x * Subs(
Derivative(f(x), x), x, 0
) + x ** 2 * Subs(Derivative(f(x), x, x), x, 0) / 2 + O(x ** 3)
assert f(x).series(x, 0, 3) == f(0) + x * Subs(
Derivative(f(x), x), x, 0
) + x ** 2 * Subs(Derivative(f(x), x, x), x, 0) / 2 + O(x ** 3)
assert f(x ** 2).series(x, 0, 3) == f(0) + x ** 2 * Subs(
Derivative(f(x), x), x, 0
) + O(x ** 3)
assert f(x ** 2 + 1).series(x, 0, 3) == f(1) + x ** 2 * Subs(
Derivative(f(x), x), x, 1
) + O(x ** 3)
class TestF(Function):
pass
assert TestF(x).series(x, 0, 3) == TestF(0) + x * Subs(
Derivative(TestF(x), x), x, 0
) + x ** 2 * Subs(Derivative(TestF(x), x, x), x, 0) / 2 + O(x ** 3)
def _find_DN(var, coeff):
x = var[0]
y = var[1]
X, Y = symbols("X, Y", integer=True)
A , B = _transformation_to_pell(var, coeff)
u = (A*Matrix([X, Y]) + B)[0]
v = (A*Matrix([X, Y]) + B)[1]
eq = x**2*coeff[x**2] + x*y*coeff[x*y] + y**2*coeff[y**2] + x*coeff[x] + y*coeff[y] + coeff[Integer(1)]
simplified = simplify(Subs(eq, (x, y), (u, v)).doit())
coeff = dict([reversed(t.as_independent(*[X, Y])) for t in simplified.args])
for term in [X**2, Y**2, Integer(1)]:
if term not in coeff.keys():
coeff[term] = Integer(0)
return -coeff[Y**2]/coeff[X**2], -coeff[Integer(1)]/coeff[X**2]
def test_dsolve_all_hint():
eq = f(x).diff(x)
output = dsolve(eq, hint='all')
# Match the Dummy variables:
sol1 = output['separable_Integral']
_y = sol1.lhs.args[1][0]
sol1 = output['1st_homogeneous_coeff_subs_dep_div_indep_Integral']
_u1 = sol1.rhs.args[1].args[1][0]
expected = {'Bernoulli_Integral': Eq(f(x), C1 + Integral(0, x)),
'1st_homogeneous_coeff_best': Eq(f(x), C1),
'Bernoulli': Eq(f(x), C1),
'nth_algebraic': Eq(f(x), C1),
'nth_linear_euler_eq_homogeneous': Eq(f(x), C1),
'nth_linear_constant_coeff_homogeneous': Eq(f(x), C1),
'separable': Eq(f(x), C1),
'1st_homogeneous_coeff_subs_indep_div_dep': Eq(f(x), C1),
'nth_algebraic_Integral': Eq(f(x), C1),
'1st_linear': Eq(f(x), C1),
'1st_linear_Integral': Eq(f(x), C1 + Integral(0, x)),
'1st_exact': Eq(f(x), C1),
'1st_exact_Integral': Eq(Subs(Integral(0, x) + Integral(1, _y), _y, f(x)), C1),
'lie_group': Eq(f(x), C1),
'1st_homogeneous_coeff_subs_dep_div_indep': Eq(f(x), C1),
'1st_homogeneous_coeff_subs_dep_div_indep_Integral': Eq(log(x), C1 + Integral(-1/_u1, (_u1, f(x)/x))),
'1st_power_series': Eq(f(x), C1),
'separable_Integral': Eq(Integral(1, (_y, f(x))), C1 + Integral(0, x)),
'1st_homogeneous_coeff_subs_indep_div_dep_Integral': Eq(f(x), C1),
'best': Eq(f(x), C1),
'best_hint': 'nth_algebraic',
'default': 'nth_algebraic',
'order': 1}
assert output == expected
assert dsolve(eq, hint='best') == Eq(f(x), C1)
def test_deriv1():
# These all requre derivatives evaluated at a point (issue 1620) to work.
# See issue 1525
f = Function('f')
g = Function('g')
x = Symbol('x')
assert f(g(x)).diff(x) == Derivative(g(x), x) * Subs(
Derivative(f(x), x), Tuple(x), Tuple(g(x)))
assert f(
2 * x).diff(x) == 2 * Subs(Derivative(f(x), x), Tuple(x), Tuple(2 * x))
assert (f(x)**3).diff(x) == 3 * f(x)**2 * f(x).diff(x)
assert (f(2 * x)**3).diff(x) == 6 * f(2 * x)**2 * Subs(
Derivative(f(x), x), Tuple(x), Tuple(2 * x))
assert f(2 + x).diff(x) == Subs(Derivative(f(x), x), Tuple(x),
Tuple(x + 2))
assert f(2 + 3 * x).diff(x) == 3 * Subs(Derivative(f(x), x), Tuple(x),
Tuple(3 * x + 2))
assert f(sin(x)).diff(x) == cos(x) * Subs(Derivative(f(x), x), Tuple(x),
Tuple(sin(x)))
assert f(3 * sin(x)).diff(x) == 3 * cos(x) * Subs(Derivative(
f(x), x), Tuple(x), Tuple(3 * sin(x)))
def test_subs_in_derivative():
expr = sin(x * exp(y))
u = Function('u')
v = Function('v')
assert Derivative(expr, y).subs(expr, y) == Derivative(y, y)
assert Derivative(expr, y).subs(y, x).doit() == \
Derivative(expr, y).doit().subs(y, x)
assert Derivative(f(x, y), y).subs(y, x) == Subs(Derivative(f(x, y), y), y,
x)
assert Derivative(f(x, y), y).subs(x, y) == Subs(Derivative(f(x, y), y), x,
y)
assert Derivative(f(x, y),
y).subs(y, g(x, y)) == Subs(Derivative(f(x, y), y), y,
g(x, y)).doit()
assert Derivative(f(x, y),
y).subs(x, g(x, y)) == Subs(Derivative(f(x, y), y), x,
g(x, y))
assert Derivative(f(x, y),
g(y)).subs(x, g(x,
y)) == Derivative(f(g(x, y), y), g(y))
assert Derivative(f(u(x), h(y)), h(y)).subs(h(y), g(x, y)) == \
Subs(Derivative(f(u(x), h(y)), h(y)), h(y), g(x, y)).doit()
assert Derivative(f(x, y), y).subs(y, z) == Derivative(f(x, z), z)
assert Derivative(f(x, y), y).subs(y, g(y)) == Derivative(f(x, g(y)), g(y))
assert Derivative(f(g(x), h(y)), h(y)).subs(h(y), u(y)) == \
Derivative(f(g(x), u(y)), u(y))
assert Derivative(f(x, f(x, x)), f(x, x)).subs(f, Lambda(
(x, y), x + y)) == Subs(Derivative(z + x, z), z, 2 * x)
assert Subs(Derivative(f(f(x)), x), f, cos).doit() == sin(x) * sin(cos(x))
assert Subs(Derivative(f(f(x)), f(x)), f, cos).doit() == -sin(cos(x))
# Issue 13791. No comparison (it's a long formula) but this used to raise an exception.
assert isinstance(v(x, y, u(x, y)).diff(y).diff(x).diff(y), Expr)
# This is also related to issues 13791 and 13795; issue 15190
F = Lambda((x, y), exp(2 * x + 3 * y))
abstract = f(x, f(x, x)).diff(x, 2)
concrete = F(x, F(x, x)).diff(x, 2)
assert (abstract.subs(f, F).doit() - concrete).simplify() == 0
# don't introduce a new symbol if not necessary
assert x in f(x).diff(x).subs(x, 0).atoms()
# case (4)
assert Derivative(f(x, f(x, y)), x,
y).subs(x, g(y)) == Subs(Derivative(f(x, f(x, y)), x, y),
x, g(y))
assert Derivative(f(x, x), x).subs(x, 0) == Subs(Derivative(f(x, x), x), x,
0)
# issue 15194
assert Derivative(f(y, g(x)),
(x, z)).subs(z, x) == Derivative(f(y, g(x)), (x, x))
df = f(x).diff(x)
assert df.subs(df, 1) is S.One
assert df.diff(df) is S.One
dxy = Derivative(f(x, y), x, y)
dyx = Derivative(f(x, y), y, x)
assert dxy.subs(Derivative(f(x, y), y, x), 1) is S.One
assert dxy.diff(dyx) is S.One
assert Derivative(f(x, y), x, 2, y,
3).subs(dyx, g(x, y)) == Derivative(g(x, y), x, 1, y, 2)
assert Derivative(f(x, x - y),
y).subs(x, x + y) == Subs(Derivative(f(x, x - y), y), x,
x + y)
def test_Subs2():
# this reflects a limitation of subs(), probably won't fix
assert Subs(f(x), x**2, x).doit() == f(sqrt(x))
def test_Subs():
assert Subs(1, (), ()) is S.One
# check null subs influence on hashing
assert Subs(x, y, z) != Subs(x, y, 1)
# neutral subs works
assert Subs(x, x, 1).subs(x, y).has(y)
# self mapping var/point
assert Subs(Derivative(f(x), (x, 2)), x, x).doit() == f(x).diff(x, x)
assert Subs(x, x, 0).has(x) # it's a structural answer
assert not Subs(x, x, 0).free_symbols
assert Subs(Subs(x + y, x, 2), y, 1) == Subs(x + y, (x, y), (2, 1))
assert Subs(x, (x, ), (0, )) == Subs(x, x, 0)
assert Subs(x, x, 0) == Subs(y, y, 0)
assert Subs(x, x, 0).subs(x, 1) == Subs(x, x, 0)
assert Subs(y, x, 0).subs(y, 1) == Subs(1, x, 0)
assert Subs(f(x), x, 0).doit() == f(0)
assert Subs(f(x**2), x**2, 0).doit() == f(0)
assert Subs(f(x, y, z), (x, y, z), (0, 1, 1)) != \
Subs(f(x, y, z), (x, y, z), (0, 0, 1))
assert Subs(x, y, 2).subs(x, y).doit() == 2
assert Subs(f(x, y), (x, y, z), (0, 1, 1)) != \
Subs(f(x, y) + z, (x, y, z), (0, 1, 0))
assert Subs(f(x, y), (x, y), (0, 1)).doit() == f(0, 1)
assert Subs(Subs(f(x, y), x, 0), y, 1).doit() == f(0, 1)
raises(ValueError, lambda: Subs(f(x, y), (x, y), (0, 0, 1)))
raises(ValueError, lambda: Subs(f(x, y), (x, x, y), (0, 0, 1)))
assert len(Subs(f(x, y), (x, y), (0, 1)).variables) == 2
assert Subs(f(x, y), (x, y), (0, 1)).point == Tuple(0, 1)
assert Subs(f(x), x, 0) == Subs(f(y), y, 0)
assert Subs(f(x, y), (x, y), (0, 1)) == Subs(f(x, y), (y, x), (1, 0))
assert Subs(f(x) * y, (x, y), (0, 1)) == Subs(f(y) * x, (y, x), (0, 1))
assert Subs(f(x) * y, (x, y), (1, 1)) == Subs(f(y) * x, (x, y), (1, 1))
assert Subs(f(x), x, 0).subs(x, 1).doit() == f(0)
assert Subs(f(x), x, y).subs(y, 0) == Subs(f(x), x, 0)
assert Subs(y * f(x), x, y).subs(y, 2) == Subs(2 * f(x), x, 2)
assert (2 * Subs(f(x), x, 0)).subs(Subs(f(x), x, 0), y) == 2 * y
assert Subs(f(x), x, 0).free_symbols == set([])
assert Subs(f(x, y), x, z).free_symbols == {y, z}
assert Subs(f(x).diff(x), x, 0).doit(), Subs(f(x).diff(x), x, 0)
assert Subs(1 + f(x).diff(x), x, 0).doit(), 1 + Subs(f(x).diff(x), x, 0)
assert Subs(y*f(x, y).diff(x), (x, y), (0, 2)).doit() == \
2*Subs(Derivative(f(x, 2), x), x, 0)
assert Subs(y**2 * f(x), x, 0).diff(y) == 2 * y * f(0)
e = Subs(y**2 * f(x), x, y)
assert e.diff(y) == e.doit().diff(
y) == y**2 * Derivative(f(y), y) + 2 * y * f(y)
assert Subs(f(x), x, 0) + Subs(f(x), x, 0) == 2 * Subs(f(x), x, 0)
e1 = Subs(z * f(x), x, 1)
e2 = Subs(z * f(y), y, 1)
assert e1 + e2 == 2 * e1
assert e1.__hash__() == e2.__hash__()
assert Subs(z * f(x + 1), x, 1) not in [e1, e2]
assert Derivative(f(x), x).subs(x, g(x)) == Derivative(f(g(x)), g(x))
assert Derivative(f(x), x).subs(x, x + y) == Subs(Derivative(f(x), x), x,
x + y)
assert Subs(f(x)*cos(y) + z, (x, y), (0, pi/3)).n(2) == \
Subs(f(x)*cos(y) + z, (x, y), (0, pi/3)).evalf(2) == \
z + Rational('1/2').n(2)*f(0)
assert f(x).diff(x).subs(x, 0).subs(x, y) == f(x).diff(x).subs(x, 0)
assert (x * f(x).diff(x).subs(x, 0)).subs(
x, y) == y * f(x).diff(x).subs(x, 0)
assert Subs(Derivative(g(x)**2, g(x), x), g(x),
exp(x)).doit() == 2 * exp(x)
assert Subs(Derivative(g(x)**2, g(x), x), g(x),
exp(x)).doit(deep=False) == 2 * Derivative(exp(x), x)
assert Derivative(
f(x, g(x)),
x).doit() == Derivative(f(x, g(x)), g(x)) * Derivative(g(x), x) + Subs(
Derivative(f(y, g(x)), y), y, x)
