How do I compute the derivative of an array in python

Question

How do I compute the derivative of an array, y (say), with respect to another array, x (say) - both arrays from a certain experiment?

e.g.

y = [1,2,3,4,4,5,6] and x = [.1,.2,.5,.6,.7,.8,.9];

I want to get dy/dx!

Answer 1

Use numpy.diff

If dx is constant

from numpy import diff
dx = 0.1
y = [1, 2, 3, 4, 4, 5, 6]
dy = diff(y)/dx
print dy 
array([ 10.,  10.,  10.,   0.,  10.,  10.])

dx is not constant (your example)

from numpy import diff
x = [.1, .2, .5, .6, .7, .8, .9]
y = [1, 2, 3, 4, 4, 5, 6]
dydx = diff(y)/diff(x)
print dydx 
[10., 3.33333,  10. ,   0. , 10. ,  10.]

Note that this approximated "derivative" has size n-1 where n is your array/list size.

Don't know what you are trying to achieve but here are some ideas:

• If you are trying to make numerical differentiation maybe finite differences formulation might help you better.
• The solution above is like a first-order accuracy approximation for the forward schema of finite differences with a non-uniform grid/array.

Answer 2

use numpy.gradient()

Please be aware that there are more advanced way to calculate the numerical derivative than simply using diff. I would suggest to use numpy.gradient, like in this example.

import numpy as np
from matplotlib import pyplot as plt

# we sample a sin(x) function
dx = np.pi/10
x = np.arange(0,2*np.pi,np.pi/10)

# we calculate the derivative, with np.gradient
plt.plot(x,np.gradient(np.sin(x), dx), '-*', label='approx')

# we compare it with the exact first derivative, i.e. cos(x)
plt.plot(x,np.cos(x), label='exact')
plt.legend()

Answer 3

I'm assuming this is what you meant:

>>> from __future__ import division
>>> x = [.1,.2,.5,.6,.7,.8,.9]
>>> y = [1,2,3,4,4,5,6]
>>> from itertools import izip
>>> def pairwise(iterable): # question 5389507
...     "s -> (s0,s1), (s2,s3), (s4, s5), ..."
...     a = iter(iterable)
...     return izip(a, a)
... 
>>> for ((a, b), (c, d)) in zip(pairwise(x), pairwise(y)):
...   print (d - c) / (b - a)
... 
10.0
10.0
10.0
>>>

question 5389507 link

That is, define dx as the difference between adjacent elements in x.

Answer 4

numpy.diff(x) computes

the difference between adjacent elements in x

just like in the answer by @tsm. As a result you get an array which is 1 element shorter than the original one. This of course makes sense, as you can only start computing the differences from the first index (1 "history element" is needed).

>>> x = [1,3,4,6,7,8]
>>> dx = numpy.diff(x)
>>> dx
array([2, 1, 2, 1, 1])

>>> y = [1,2,4,2,3,1]
>>> dy = numpy.diff(y)
>>> dy
array([ 1,  2, -2,  1, -2])

Now you can divide those 2 resulting arrays to get the desired derivative.

>>> d = dy / dx
>>> d
array([ 0.5,  2. , -1. ,  1. , -2. ])

If for some reason, you need a relative (to the y-values) growth, you can do it the following way:

>>> d / y[:-1]
array([ 0.5       ,  1.        , -0.25      ,  0.5       , -0.66666667])

Interpret as 50% growth, 100% growth, -25% growth, etc.

Full code:

import numpy
x = [1,3,4,6,7,8]
y = [1,2,4,2,3,1]
dx = numpy.diff(x)
dy = numpy.diff(y)
d = dy/dx