Ugh. How do you even nest chain rule derivatives? How do you even nest chain rule into other rules for that matter?
This is the last question on this worksheet:
(x2-2)4(3x+4)5
Heeeellllpppp.
Welp, let's start with the basics of a chain rule, using Leibnitz notation since it's easier to visualize here:
Dy Dy Du
-- = -- * --
Dx Du Dx
Now, to nest these together, you can just apply it to the bits inside:
Dy Dy Dv
-- = -- * --
Du Dv Du
Dy ┌ Dy Dv ┐ Du
-- = │ -- * -- │ * --
Dx └ Dv Du ┘ Dx
So, for example, if you did an equation
y =
e5x2, you'd have:
u(x) = 5x2
y(u) = eu (just substitute u in by using the above)
Dy/Du = eu
Du/Dx = 10x
Dy/Dx = Dy/Du * Du/Dx = eu * 10x = 10x * e5x2
However, in this case, you're not really nesting the chain rule. It's much easier to look at your function as the product of two functions multiplied together, where your two functions are (x
2-2)
4 and (3x+4)
5. So, I suppose the questions first are:
(a) Have you learned the product rule for derivatives, and
(b) Are you allowed to use it for this function, or are the instructions so very specific about the intended method you're supposed to use that you must use the chain rule?
EDIT:
Oh, I think I might see what you're asking. Are you just having trouble using the chain rule on each of these component functions? It might help you out just to treat each one as its own problem. Take the derivative of one of the component functions using the chain rule, then take the derivative of the other function using the chain rule, then plug each of those derivatives into the product rule to come out with your final answer.
So, to elaborate, you've got your function
f =
g *
h. To find
f' (feeling lazy, so now it's Lagrangian notation), find
g' and
h' by whatever means necessary (the chain rule, here, like the other problems on your worksheet), then use those in the product rule (
g *
h)' =
g' *
h +
g *
h'. No special tricks, but feel free to point out where you're having trouble if there's an issue.
EDIT 2: Glitched out on some of the notation. Should be corrected now. Also elaborated on the edit a bit.