2.3 KiB
| title |
|---|
| Chain Rule Introduction |
Chain Rule Introduction
Chain Rule is used to compute the derivative of a composition of functions.
Let F be a real valued function which is a composite of two functions f and g i.e. F(x) = f(g(x))and both f(x) and g(x) are differentiable.
Let the derivative D{F(x)} is denoted as F'(x).
By Chain Rule,
F'(x) = f'(g(x)).g'(x)
Suppose, g(x) = t then F(x) = f(g(x)) can be rewritten as F(x)=f(t) then in Leibniz's notation Chain Rule can be rewritten as :
d(F)/dx = df/dt . dt/dx
Example 1. To compute derivative of sin(ax+b)
Solution : The function can be visualized as a composite of two functions. F(x)= f(g(x))
t= g(x)= ax+b and f(t)=sin(t)
f(t)=sin(t) => df/dt = cos(t)
t= g(x) = ax+b => dt/dx = a
Now by Chain Rule:
d(F)/dx = df/dt . dt/dx
=> d(F)/dx = a . cost(t) = a.cos(ax+b)
OR
We can directly apply the formula F'(x) = f'(g(x)).g'(x) = cos(ax+b) . a
For a function composite of more than two function :
Let F be a real valued function which is a composite of four functions r s t u i.e. F(x)=r(s(t(u(x)))) and all the functions r(x) s(x) t(x) u(x) are differentiable.
Let the derivative D{F(x)} is denoted as F'(x).
By Chain Rule,
F'(x) = r'(s(t(u(x)))).s'(t(u(x))).t'(u(x)).u'(x)
Suppose, a = u(x), b = t(a), c = s(b) then F(x)=r(s(t(u(x)))) can be re-written as F(x)=r(c)
then, F(x)=r(c) => d(F)/dx = dr/dc . dc/dx ___ (eqn 1)
c = s(b) => dc/dx = ds/db . db/dx ___ (eqn 2)
b = t(a) => db/dx = dt/da . da/dx ___ (eqn 3)
a = u(x) => da/dx = du/dx ___ (eqn 4)
Putting the value of eqn 2 3 4 in eqn 1, we will get :
d(F)/dx = dr/dc . ds/db . dt/da . du/dx
Example 2. To compute derivative of sin(cos((mx+n)^3))
Solution : The function can be visualized as a composite of four functions. F(x)= r(s(t(u(x))))
where a = u(x) = mx+n
b = t(a) = a^3
c = s(b) = cos(b) then F(x)=r(s(t(u(x)))) can be re-written as F(x)= r(c) =sin(c)
Now, By chain rule : d(F)/dx = dr/dc . ds/db . dt/da . du/dx
=> d(F)/dx = cos(c) . -sin(b) . 3a^2 . m
=> d(F)/dx = cos(cos((mx+n)^3)) . -sin((mx+n)^3)) . 3(mx+n)^2 . m
OR
We can directly apply the formula,
F'(x) = r'(s(t(u(x)))).s'(t(u(x))).t'(u(x)).u'(x) = cos(cos((mx+n)^3)) . -sin((mx+n)^3)) . 3(mx+n)^2 . m