The chain rule deals with functions composed of other functions, where a change in the inner function affects the outer function. It is used when taking the derivative of a function of a function, such as y=f(g(x)). The chain rule states that the derivative of the outer function is the derivative of the outer function with respect to the inner function, multiplied by the derivative of the inner function. Some examples demonstrate applying the chain rule to find the derivative of various functions.

1. By
Trevor Bennett

2.  The chain rule basically deals with functions
within a function. There is a strong
relationship between them. If you consider
the links on a chain, one link overlaps the
other.
 If there is a change to either the inside or
outside link, it affect the relationship between
them.

3.  I believe so far that you have covered
derivatives of functions. For example y =f(x)
and dy/dx = f’(x).
 But now considertwo functions f(x) and g(x).
We will loo at a function within a function.
 For instance y = f(g(x)). So then what is y’?
 The chain rule is what is applied!!

4.  If y = f(g(x))
 y’= f ’(g(x)) . g’(x) OR

 Alternatively
 You can let u = g(x) and so it becomes
 y = f(u)
 And then to determine y’, we get
 y’ = dy/dx = dy/du * du/dx.

5.  It can also be written as
 y’ = dy/dx = du/dx * dy/du

6.  A simple example could be
 y = (2x +5)2
 Let u = 2x +5 therefore y =u2
 du/dx = 2 + 0 =2
 dy/du = 2u1 = 2(2x +5)
 Therefore dy/dx = dy/du *du/dx
 = 2*2(2x +5)=4(2x + 5)
 = 8x + 20

7.  y = 5/(3x-5)3
 y’ = 5(3x-5)-3
 y’ =-3*5(3x-5)-4 *(3)
 y’ =-45(3x-5)-4
 y’ =-45/(3x-5)4

8. 
 y = sin(x7)
 let u = x7
 y’ = dy/dx = dy/du *du/dx
 dy/du = cos(u) & du/dx = 7x6
 y’ = cos(x7) *7x6
 y’ =7x6*cos(x7)