chain Rule of partial derivitives

I realize that i said that i want this blog to be allabout linear algebra but i feel that that limits this blog alot. If your up for a little advanced calculus please sit bak relax and enjoy my chain rule.

So if you remeber chain rule from your first year maths course this will be very similer and i hope that we dont get too tripped up. Before we actually begin that let’s first review the notation for the chain rule for functions of one variable.

The notation that’s probably familiar to most people is the following.

There is an alternate notation however that while probably not used much in Calculus I is more convenient at this point because it will match up with the notation that we are going to be using in this section.  Here it is.

Notice that the derivative  really does make sense here since if we were to plug in for x then y really would be a function of t.  One way to remember this form of the chain rule is to note that if we think of the two derivatives on the right side as fractions the dx’s will cancel to get the same derivative on both sides. 

Wow that was a mouth full and haven't even began yet. if your need 5 miniuts. are you good ? ok lets go. at the end of this post there will be a pictre of a chained potato, uts try to make it to the end for that potato it needs you.

Lets look at a sample problem,  

Where :

,  

and we try to compute

This case is analogous to the standard chain rule from Calculus I that we looked at above.  In this case we are going to compute an ordinary derivative since z really would be a function of t only if we were to substitute in for x and y.

The chain rule for this case is,

So, basically what we’re doing here is differentiating f with respect to each variable in it and then multiplying each of these by the derivative of that variable with respect to t.  The final step is to then add all this up.

ill leave finding examples to your discression. if you would like me to post practice problems ill be happy to just let me know in the discussion below. 

As promised your chained potato!!!