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title: Chain Rule Introduction
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# 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