Calculating the derivative of points in python

I want to calculate the derivative of points, a few internet posts suggested using np.diff function. However, I tried using np.diff against manually calculated results (chose a random polynomial equation and differentiated it) to see if I end up with the same results. I used the following eq : Y = (X^3) + (X^2) + 7 and the results i ended up with were different. Any ideas why?. Is there any other method to calculate the differential.

In the problem, I am trying to solve, I have recieved data points of fitted spline function ( not the original data that need to be fitted by spline but the points of the already fitted spline). The x-values are at equal intervals. I only have the points and no equation, what I require is to calculate, the first, second and third derivatives. i.e dy/dx, d2y/dx2, d3y/dx3. Any ideas on how to do this?. Thanks in advance.

xval = [1,2,3,4,5]
yval = []
yval_dashList = []

#selected a polynomial equation
def calc_Y(X):
      Y = (X**3) + (X**2) + 7
      return(Y)

#calculate y values using equatuion 
for i in xval:
    yval.append(calc_Y(i))

#output: yval = [9,19,43,87,157]

#manually differentiated the equation or use sympy library (sym.diff(x**3 + x**2 + 7))
def calc_diffY(X):
   yval_dash = 3*(X**2) + 2**X

#store differentiated y-values in a list
for i in xval:
    yval_dashList.append(yval_dash(i))

#output: yval_dashList = [5,16,35,64,107]

#use numpy diff method on the y values(yval)
numpyDiff = np.diff(yval)
#output: [10,24,44,60]

The values of numpy diff method [10,24,44,60] is different from yval_dashList = [5,16,35,64,107]