Mehrdad - 1 year ago 69
Python Question

# How to find the nth derivative given the first derivative with SymPy?

Given some f and the differential equation x'(t) = f(x(t)), how do I compute x(n)(t) in terms of x(t)?

For example, given f(x(t)) = sin(x(t)),
I want to obtain x(3)(t) = (cos(x(t))2 − sin(x(t))2) sin(x(t)).

So far I've tried

``````>>> from sympy import diff, sin
>>> from sympy.abc import x, t
>>> diff(sin(x(t)), t, 2)
``````

which gives me

``````-sin(x(t))*Derivative(x(t), t)**2 + cos(x(t))*Derivative(x(t), t, t)
``````

but I'm not sure how to tell SymPy what
`Derivative(x(t), t)`
is and have it figure out
`Derivative(x(t), t, t)`
, etc. automatically.

``````def diff(x_derivs_known, t, k, simplify=False):
try: n = len(x_derivs_known)
except TypeError: n = None
if n is None:
result = sympy.diff(x_derivs_known, t, k)
if simplify: result = result.simplify()
elif k < n:
result = x_derivs_known[k]
else:
i = n - 1
result = x_derivs_known[i]
while i < k:
result = result.diff(t)
j = len(x_derivs_known)
x0 = None
while j > 1:
j -= 1
result = result.subs(sympy.Derivative(x_derivs_known[0], t, j), x_derivs_known[j])
i += 1
if simplify: result = result.simplify()
return result
``````

Example:

``````>>> diff((x(t), sympy.sin(x(t))), t, 3, True)
sin(x(t))*cos(2*x(t))
``````

Here is one approach that returns a list of all derivatives up to `n`-th order

``````import sympy as sp

x = sp.Function('x')
t = sp.symbols('t')

f = lambda x: x**2 #sp.exp, sp.sin
n = 4 #3, 4, 5

deriv_list = [x(t), f(x(t))]  # list of derivatives [x(t), x'(t), x''(t),...]
for i in range(1,n):
df_i = deriv_list[-1].diff(t).replace(sp.Derivative,lambda *args: f(x(t)))
deriv_list.append(df_i)

print(deriv_list)
``````

`[x(t), x(t)**2, 2*x(t)**3, 6*x(t)**4, 24*x(t)**5]`

With `f=sp.sin` it returns

`````` [x(t), sin(x(t)), sin(x(t))*cos(x(t)), -sin(x(t))**3 + sin(x(t))*cos(x(t))**2, -5*sin(x(t))**3*cos(x(t)) + sin(x(t))*cos(x(t))**3]
``````

EDIT: A recursive function for the computation of the `n`-th derivative:

``````def der_xt(f, n):
if n==1:
return f(x(t))
else:
return der_xt(f,n-1).diff(t).replace(sp.Derivative,lambda *args: f(x(t)))

print(der_xt(sp.sin,3))
``````

`-sin(x(t))**3 + sin(x(t))*cos(x(t))**2`

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