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Chain Rule
Suppose that are differentiable. And The derivative of compose of and , is:
Formula:
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Graphical Representation:
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Usage:
ChainRule( f(u), g(x), x)
f(u) : function of
g(x) : function of
x : variable
Author: chu-ching huang
Chang-gung University, Taiwan
cchuang_AT_mail.cgu.edu.tw
============ 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. ==========
============ 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. ==========
============ 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. ==========
============ 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. ==========
============ Chain Rule =============
The derivative of f(g(x)) = sin(x**3)
with f(u)=sin(u)
u=g(x)=x**3
==>
========= .oo000oo. End .oo000oo. ==========