CoCalc -- 2001.sagews

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Chain Rule

Suppose that $f, g$ are differentiable. And The derivative of compose of $f$ and $g$ , $F=f(g(x))$ is:

Formula:

$\frac{d}{d x}F(x)$	=	$\frac{d}{d x} f(g(x)))$
	=	$\frac{d}{d g(x)}$ $f(g(x))$ • $\frac{ dg(x)}{d x}$
	=	$f'(u) • u'(x)$
	=	$f'(g(x)) g'(x)$

Graphical Representation:

	$g(x)$		$f(")$
$x$	$\longrightarrow$	$g(x)$	$\longrightarrow$	$f(g(x))$
	$\downarrow$		$\downarrow$
	$g'(x)$	$\times$	$f'(")$
		$\downarrow$ $f'(g(x))g'(x)$

Usage:

ChainRule( f(u), g(x), x)

f(u) : function of

u=g(x)

g(x) : function of

x

x : variable

x

Author: chu-ching huang

Chang-gung University, Taiwan

cchuang_AT_mail.cgu.edu.tw

# -*- coding: utf-8 -*-
from sympy import *
var('u x')
def ChainRule(f,g,x):
    f=f
    g=g
    F=f.subs(u,g)
    html("============ <span style=\"color: #0000ff;\"><strong> Chain Rule (鍊鎖法則) </strong></span>=============")
    html('The derivative of \
    <span style=\"color: #ff0000;\"><strong>$f(g(x))$</strong></span> \
    =\
    <span style=\"color: #335500;\"><strong>$%s$</strong></span>\
    '%F)
    print '\nwith  \t f(u)=%s \n     \t u=g(x)=%s \nis' %(f,g)
    html('\t<span style=\"color: #336600;\"><strong>$f\'(g(x)) = %s$</strong></span>'%diff(F,x))
    html('\nsince \
    <span style=\"color: #ff0000;\"><strong>f\'(u)=%s</strong></span> \
     and \
    <span style=\"color: #ff0000;\"><strong> u\'(x)=%s</strong></span> \
    '%(diff(f,u),diff(g,x)))
    print '\n==>\n'
    print ' dF(x)/dx \t= df(g(x))/dx = d(%s)/dx \n \
    \t\t= df(u)/dx = d(%s)/dx\n \
    \t\t= df(u)/du*du/dx \n\
    \t\t= f\'(u) * u\'(x) = [%s]*[%s] \n \t\t= [%s]*[%s] \n   \t\t= %s'\
               %(F,f,diff(f,u),diff(g,x),diff(f,u).subs(u,g),diff(g,x),diff(F,x))
    print "\nThe derivative of F(x) is:\n" 
    pretty_print(diff(F,x))
    print "\n"
    html('========= <span style=\"color: #0000ff;\"><strong> .oo000oo. End .oo000oo. </strong></span> ==========')

var('n')
ChainRule(u^n,sin(x),x)

============ Chain Rule (鍊鎖法則) =============

The derivative of f(g(x)) = sin(x)**n

with  	 f(u)=u**n 
     	 u=g(x)=sin(x) 
is

f'(g(x)) = n*sin(x)**(-1 + n)*cos(x)

since f'(u)=n*u**(-1 + n) and u'(x)=cos(x)

==>

 dF(x)/dx 	= df(g(x))/dx = d(sin(x)**n)/dx 
     		= df(u)/dx = d(u**n)/dx
     		= df(u)/du*du/dx 
    		= f'(u) * u'(x) = [n*u**(-1 + n)]*[cos(x)] 
 		= [n*sin(x)**(-1 + n)]*[cos(x)] 
   		= n*sin(x)**(-1 + n)*cos(x)

The derivative of F(x) is:

     -1 + n          
n⋅sin      (x)⋅cos(x)

========= .oo000oo. End .oo000oo. ==========

ChainRule(cos(u),x**3,x)

============ Chain Rule (鍊鎖法則) =============

The derivative of f(g(x)) = cos(x**3)

with  	 f(u)=cos(u) 
     	 u=g(x)=x**3 
is

f'(g(x)) = -3*x**2*sin(x**3)

since f'(u)=-sin(u) and u'(x)=3*x**2

==>

 dF(x)/dx 	= df(g(x))/dx = d(cos(x**3))/dx 
     		= df(u)/dx = d(cos(u))/dx
     		= df(u)/du*du/dx 
    		= f'(u) * u'(x) = [-sin(u)]*[3*x**2] 
 		= [-sin(x**3)]*[3*x**2] 
   		= -3*x**2*sin(x**3)

The derivative of F(x) is:

    2    ⎛ 3⎞
-3⋅x ⋅sin⎝x ⎠

========= .oo000oo. End .oo000oo. ==========

ChainRule(u**3,cos(x),x)

============ Chain Rule (鍊鎖法則) =============

The derivative of f(g(x)) = cos(x)**3

with  	 f(u)=u**3 
     	 u=g(x)=cos(x) 
is

f'(g(x)) = -3*cos(x)**2*sin(x)

since f'(u)=3*u**2 and u'(x)=-sin(x)

==>

 dF(x)/dx 	= df(g(x))/dx = d(cos(x)**3)/dx 
     		= df(u)/dx = d(u**3)/dx
     		= df(u)/du*du/dx 
    		= f'(u) * u'(x) = [3*u**2]*[-sin(x)] 
 		= [3*cos(x)**2]*[-sin(x)] 
   		= -3*cos(x)**2*sin(x)

The derivative of F(x) is:

      2          
-3⋅cos (x)⋅sin(x)

========= .oo000oo. End .oo000oo. ==========

def ChainRule2(f,g,x):
    f=f
    g=g
    e=''
    F=f.subs(u,g)
    
    html("============ <span style=\"color: #0000ff;\"><strong> Chain Rule ('鍊鎖法則') </strong></span>=============")
    html('The derivative of \
    <span style=\"color: #ff0000;\"><strong>$f(g(x))$</strong></span> \
    =\
    <span style=\"color: #335500;\"><strong>$%s$</strong></span>\
    '%F)
    print '\n  with  \t f(u)=%s \n \t\t u=g(x)=%s \n' %(f,g)
   
   
    print '\n==>'
    
    print "\n"
    html('\t\t\t$%s$\t %10s  \t$%s$' %(g,e,f))
    html('\t\t $x$ \t $\longrightarrow$ \t $%s$ \t $\longrightarrow$ \t $%s$' %(g,f.subs(u,g)))   
    html('\t(<span style=\"color: #ff0000;\">diff</span>)\t\t  $ \downarrow$\t\t\t   $\downarrow$')

    html('<span style=\"color: #ff0000;\">\t\t\t$%10s$\t        \t$%10s$  </span>' %(g.diff(x),f.diff(u)))
    html('<span style=\"color: #336600;\">\t($u=%s$)\t\t  $\downarrow$</span>' %g)
    html('<span style=\"color: #0000ff;\">\t\t\t  $%s$*%s$</span>' %(g.diff(x),f.diff(u).subs(u,g)))
    html('\n========= <span style=\"color: #0000ff;\"><strong> .oo000oo. End .oo000oo. </strong></span> ==========')

ChainRule2(exp(u),-x^2,x)

============ Chain Rule ('鍊鎖法則') =============

The derivative of f(g(x)) = exp(-x**2)

with  	 f(u)=exp(u) 
 		 u=g(x)=-x**2 


==>

-x**2 exp(u)

x \longrightarrow -x**2 \longrightarrow exp(-x**2)

(diff) \downarrow \downarrow

-2*x exp(u)

(u=-x**2) \downarrow

-2*x*exp(-x**2)

========= .oo000oo. End .oo000oo. ==========

def ChainRule3(f,g,x):
    f=f
    g=g
    e=''
    F=f.subs(u,g)
    
    html("============ <span style=\"color: #0000ff;\"><strong> Chain Rule  </strong></span>=============")
    html('The derivative of \
    <span style=\"color: #ff0000;\"><strong>$f(g(x))$</strong></span> \
    =\
    <span style=\"color: #335500;\"><strong>$%s$</strong></span>\
    '%F)
    print '\n  with  \t f(u)=%s \n \t\t u=g(x)=%s \n' %(f,g)
   
   
    print '\n==>'
    
    print "\n"
    html('<table style=\"width: 476px; height: 129px;\" border=\"0\">')
    html('<tbody>')
    html('<tr>')
    html('<td></td><td></td><td>$%s$</td>' %g)
    html('<td></td><td>$%s$</td>' %f)
    html('</tr>')
    html('<tr>')
    html('<td></td><td>$x$ </td><td>$\longrightarrow$</td><td> $%s$</td>' %g)
    html('<td>$\longrightarrow$</td><td>$%s$</td>' %f.subs(u,g))
    html('</tr>')
    html('<tr>')
    html('<td>(<span style=\"color: #ff0000;\">diff</span>)</td>')
    html('<td> $\downarrow$</td><td></td><td></td><td> $\downarrow$</td>')
    html('</tr>')
    html('<tr>')
    html('<td></td><td>')
    html('<td>$%s$</td><td>  $\\times$</td><td></td><td>$%s$</td>' %(g.diff(x),f.diff(u)))
    html('</tr>')
    html('<tr>')
    html('<td></td><td></td><td></td><td>  $\downarrow$</td>')
    html('</tr>')
    html('<tr>')
    html('<td>($u=%s$)</td><td></td><td></td>' %g)
    html('<td>$%s$ $%s$</td>' %(g.diff(x),f.diff(u).subs(u,g)))
    html('</tr>')
    html('</tbody>')
    html('</table>')
   
    html('\n========= <span style=\"color: #0000ff;\"><strong> .oo000oo. End .oo000oo. </strong></span> ==========')

ChainRule3(sin(u),x^3,x)

============ Chain Rule =============

The derivative of f(g(x)) = sin(x**3)

with  	 f(u)=sin(u) 
 		 u=g(x)=x**3 


==>

x**3

sin(u)

x \longrightarrow x**3

\longrightarrowsin(x**3)

(diff)

\downarrow \downarrow

3*x**2 \timescos(u)

\downarrow

(u=x**3)

3*x**2 cos(x**3)

========= .oo000oo. End .oo000oo. ==========