Expert: Ahmed Salami - 11/30/2004

Question
what is the chain rule and can you explain how it works


Answer
Hi,
The chain rule allows you to use the normal rules of differentiation for even more complex arrangements. For instance, we know that
y = x^5,   dy/dx = 5x^4
but what if you want to evaluate dy/dx for
y = (3x + 4)^5   ?
You simply use the chain rule by letting say
u = 3x + 4
and knowing that dy/dx = (dy/du).(du/dx)
But
y = u^5 which gives us dy/du = 5u^4
and u = 3x + 4  ,    du/dx = 3
Therefore,
dy/dx = 3.5u^4
     = 3.5(3x + 4)^4
     = 15(3x + 4)^4
The rule can also be extended at time to three or more variables applied.
One example is
dy/dx = (dy/du)(du/dv)(dv/dp)(dp/dx)
I hope you get the idea and find it useful.
Good luck.
Regards.