Odvod

Primer

Sestavimo tabelo odvodov elementarnih funkcij in pravil za odvajanje. Za simbolično odvajanje lahko uporabimo funkcijo diff iz knjižnice sympy.



In [6]:

    
simplify(diff(x**n,x))









    Out[6]:





$$n x^{n - 1}$$



In [1]:

    
from sympy import *
init_printing()
x,n = symbols('x n')
funkcije = [1,x**n,sin(x),cos(x), exp(x),log(x)]
tabela = [[f,diff(f,x)] for f in funkcije]
tabela









    Out[1]:





$$\left [ \left [ 1, \quad 0\right ], \quad \left [ x^{n}, \quad \frac{n x^{n}}{x}\right ], \quad \left [ \sin{\left (x \right )}, \quad \cos{\left (x \right )}\right ], \quad \left [ \cos{\left (x \right )}, \quad - \sin{\left (x \right )}\right ], \quad \left [ e^{x}, \quad e^{x}\right ], \quad \left [ \log{\left (x \right )}, \quad \frac{1}{x}\right ]\right ]$$

Lepši izpis tabele dobimo, če uporabimo knjižnico za delo s tabelami in podatki Pandas.



In [2]:

    
from pandas import DataFrame
DataFrame(tabela,columns={"$f(x)$","$f'(x)$"})









    Out[2]:






  
    
      
      $f'(x)$
      $f(x)$
    
  
  
    
      0
      1
      0
    
    
      1
      x**n
      n*x**n/x
    
    
      2
      sin(x)
      cos(x)
    
    
      3
      cos(x)
      -sin(x)
    
    
      4
      exp(x)
      exp(x)
    
    
      5
      log(x)
      1/x



In [3]:

    
# za lepši izpis uporabimo funkcijo latex
tabela =[['$$%s$$' % latex(f),'$$%s$$'% latex(diff(f,x))] for f in funkcije ]
DataFrame(tabela,columns={"funkcija $f(x)$","odvod $f'(x)$"})









    Out[3]:






  
    
      
      odvod $f'(x)$
      funkcija $f(x)$
    
  
  
    
      0
      $$1$$
      $$0$$
    
    
      1
      $$x^{n}$$
      $$\frac{n x^{n}}{x}$$
    
    
      2
      $$\sin{\left (x \right )}$$
      $$\cos{\left (x \right )}$$
    
    
      3
      $$\cos{\left (x \right )}$$
      $$- \sin{\left (x \right )}$$
    
    
      4
      $$e^{x}$$
      $$e^{x}$$
    
    
      5
      $$\log{\left (x \right )}$$
      $$\frac{1}{x}$$

Pravila za odvajanje

Sestavimo še tabelo pravil za odvajanje. Vključimo pravila za osnovne operacije in kompozitum.



In [4]:

    
%%javascript
MathJax.Hub.Config({
    "HTML-CSS": { linebreaks: {automatic: false } }
  });
// preprečimo prelom vrstic v tabeli



In [5]:

    
f,g = symbols("f,g")
import pandas as pd
pd.options.display.max_colwidth=1000 

operacije = [f(x)+g(x), f(x)*g(x), f(x)/g(x), f(x)**g(x),f(g(x))]
fmt = "$$%s$$"
tabela = [[fmt % latex(op),fmt % latex(simplify(diff(op,x)))] for op in operacije]
df_tabela = DataFrame(tabela, columns=["funkcija $f(x)$","pravilo za odvod $f'(x)$"],) 
df_tabela









    Out[5]:






  
    
      
      funkcija $f(x)$
      pravilo za odvod $f'(x)$
    
  
  
    
      0
      $$f{\left (x \right )} + g{\left (x \right )}$$
      $$\frac{d}{d x} f{\left (x \right )} + \frac{d}{d x} g{\left (x \right )}$$
    
    
      1
      $$f{\left (x \right )} g{\left (x \right )}$$
      $$f{\left (x \right )} \frac{d}{d x} g{\left (x \right )} + g{\left (x \right )} \frac{d}{d x} f{\left (x \right )}$$
    
    
      2
      $$\frac{f{\left (x \right )}}{g{\left (x \right )}}$$
      $$\frac{1}{g^{2}{\left (x \right )}} \left(- f{\left (x \right )} \frac{d}{d x} g{\left (x \right )} + g{\left (x \right )} \frac{d}{d x} f{\left (x \right )}\right)$$
    
    
      3
      $$f^{g{\left (x \right )}}{\left (x \right )}$$
      $$\left(f{\left (x \right )} \log{\left (f{\left (x \right )} \right )} \frac{d}{d x} g{\left (x \right )} + g{\left (x \right )} \frac{d}{d x} f{\left (x \right )}\right) f^{g{\left (x \right )} - 1}{\left (x \right )}$$
    
    
      4
      $$f{\left (g{\left (x \right )} \right )}$$
      $$\frac{d}{d x} g{\left (x \right )} \left. \frac{d}{d \xi_{1}} f{\left (\xi_{1} \right )} \right|_{\substack{ \xi_{1}=g{\left (x \right )} }}$$

Naloga

Izpiši tabelo odvodov funkcij $\tan$ in inverznih trigonometričnih funkcij $\arcsin$, $\arccos$ in $\arctan$. Nato izpiši še pravilo za odvod $$\frac{1}{f(x)}\text{ in }\log(f(x)).$$

>> naprej: tangenta



In [55]:

    
import disqus
%reload_ext disqus
%disqus matpy

	$f'(x)$	$f(x)$
0	1	0
1	x**n	nx*n/x
2	sin(x)	cos(x)
3	cos(x)	-sin(x)
4	exp(x)	exp(x)
5	log(x)	1/x

	odvod $f'(x)$	funkcija $f(x)$
0	$$1$$	$$0$$
1	$$x^{n}$$	$$\frac{n x^{n}}{x}$$
2	$$\sin{\left (x \right )}$$	$$\cos{\left (x \right )}$$
3	$$\cos{\left (x \right )}$$	$$- \sin{\left (x \right )}$$
4	$$e^{x}$$	$$e^{x}$$
5	$$\log{\left (x \right )}$$	$$\frac{1}{x}$$

	funkcija $f(x)$	pravilo za odvod $f'(x)$
0	$$f{\left (x \right )} + g{\left (x \right )}$$	$$\frac{d}{d x} f{\left (x \right )} + \frac{d}{d x} g{\left (x \right )}$$
1	$$f{\left (x \right )} g{\left (x \right )}$$	$$f{\left (x \right )} \frac{d}{d x} g{\left (x \right )} + g{\left (x \right )} \frac{d}{d x} f{\left (x \right )}$$
2	$$\frac{f{\left (x \right )}}{g{\left (x \right )}}$$	$$\frac{1}{g^{2}{\left (x \right )}} \left(- f{\left (x \right )} \frac{d}{d x} g{\left (x \right )} + g{\left (x \right )} \frac{d}{d x} f{\left (x \right )}\right)$$
3	$$f^{g{\left (x \right )}}{\left (x \right )}$$	$$\left(f{\left (x \right )} \log{\left (f{\left (x \right )} \right )} \frac{d}{d x} g{\left (x \right )} + g{\left (x \right )} \frac{d}{d x} f{\left (x \right )}\right) f^{g{\left (x \right )} - 1}{\left (x \right )}$$
4	$$f{\left (g{\left (x \right )} \right )}$$	$$\frac{d}{d x} g{\left (x \right )} \left. \frac{d}{d \xi_{1}} f{\left (\xi_{1} \right )} \right\|_{\substack{ \xi_{1}=g{\left (x \right )} }}$$