定义一个作为函数导数的函数

Question

I was wondering if is there is a way to define a function that is a derivative of a function. I'm new to python so I don't no much, I tired looking up stuff that might be similar but nothing has worked so far. This is what I have for my code right now.

import sympy as sp

import math 

x = sp.Symbol('x')


W = 15 #kN/m
E = 70 # Gpa
I = 52.9*10**(-6) #m**4
L = 3 #m 

e = 0.01
xi = 1.8 
y = 9


def f(x):
    return ( ( y*3*(math.pi**4)*E*I/(W*L) ) - ( 48*(L**3)*math.cos(math.pi*x/(2*L)) ) + ( 48*(L**3) ) + ( (math.pi**3)*(x**3) ) )/(3*L*(math.pi**3))**(1/2)

def derv(f,x):
    return sp.diff(f)

print (derv(f,x))

Also, I don't understand what x = sp.Symbol('x') does, so if someone could explain that, that would be awesome.

Any help is appreciated.

Answer 1

You are conflating two different things: python functions like f and math functions, which you can express with sympy like y = π * x/3 . f is a python function that returns a sympy expression. sympy lets you stay in the world of symbolic math functions by defining variables like x = sp.Symbol('x') So calling f() produces a symbolic math function like:

You can use sympy to find the derivative of the symbolic function returned by f() but you need to define it with the sympy versions of the cos() function (and sp.pi if you want to keep it symbolic).

For example:

import sympy as sp
    
x = sp.Symbol('x')
   
W = 15 #kN/m
E = 70 # Gpa
I = 52.9*10**(-6) #m**4
L = 3 #m 

e = 0.01
xi = 1.8 
y = 9


def f(x):
    return ( ( y*3*(sp.pi**4)*E*I/(W*L) ) - ( 48*(L**3)*sp.cos(sp.pi*x/(2*L)) ) + ( 48*(L**3) ) + ( (sp.pi**3)*(x**3) ) )/(3*L*(sp.pi**3))**(1/2)

def derv(f,x):
    return sp.diff(f(x)) # pass the result of f() which is a sympy function

derv(f,x)

Answer 2

You've programmed the function. it appears to be a simple function of two independent variables x and y.

Could be that x = sp.Symbol('x') is how SymPy defines the independent variable x. I don't know if you need one or another one for y.

You know enough about calculus to know that you need a derivative. Do you know how to differentiate a function of a single independent variable? It helps to know the answer before you start coding.

y*3*(math.pi**4)*E*I/(W*L) ) - ( 48*(L**3)*math.cos(math.pi*x/(2*L)) ) + ( 48*(L**3) ) + ( (math.pi**3)*(x**3) ) )/(3*L*(math.pi**3))**(1/2)

Looks simple.

There's only one term with y in it. The partial derivative wrt y leaves you with 3*(math.pi**4)*E*I/(W*L) )

There's only one term with C x**3 in it. That's easy to differentiate: 3 C*x**2.

What's so hard? What's the problem?

Answer 3

In traditional programming, each function you write is translated to a series of commands that are then sent to the CPU and the result of the calculation is returned. Therefore, symbolic manipulation, like what we humans do with algebra and calculus, doesn't make any sense to the computer. Sympy gets around this by overriding Python's normal arithmetic operators, allowing you to do generate algebraic functions that can be manipulated similarly to how we humans do math. That's what sp.Symbols('x') is doing: providing you with a symbolic variable you can work with (you're also naming it in sympy).

If you want to evaluate your derivative, simply call evalf with the numerical value you want to assign to x.