The problem

Sometimes we need some validation for the derivatives of a function we are taking. I had a discussion regarding this with Tushar, and he suggested using the sympy library to verify the derivatives. We can not rely entirely on these symbolic libraries for the accuracy of the solution as they are still under active development. Still, I think they can give us some confidence in the solution.

The solution

Define the derivative in symbolic language and use the diff function of sympy. In this example, my function to derivate is the p-norm of another function.

\[I = \left(\int{f(x)^p dx}\right)^{1/p}\]

from sympy import *
from IPython.display import display, Latex
x,p = symbols("x p");
f = symbols("f", cls=Function);
I = integrate(f(x)**p,x)**(1/p)
sol = simplify(I.diff(f(x)));
print(sol)
print(latex(sol))

Output

Integral(f(x)**p, x)**((1 - p)/p)*Integral(f(x)**(p - 1), x)
\left(\left(\int f^{p}{\left(x \right)}\, dx\right)^{\frac{1 - p}{p}}\right) \int f^{p - 1}{\left(x \right)}\, dx

We can then see this integral in latex by using the command.

display(Latex('$'+latex(sol)+'$'))

Output

\[\left(\left(\int f^{p}{\left(x \right)}\, dx\right)^{\frac{1 - p}{p}}\right) \int f^{p - 1}{\left(x \right)}\, dx\]