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

Question

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. Answer: Here's my final solution based on the answers I received below: ``` 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 re…

Accepted Answer

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`

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

Solution

Problem