I would probably do a logarithmic differentiation. Taking on each sides gives . I would then proceed to (implicitly) differentiate.
Can you take it from there?
Let's say we had a function like this:
I'm just wondering what the best method for differentiating this type of function. I'm thinking there is likely a trick or shortcut to it rather than making a messy chain, but I just don't know it yet.
How would you guys and gals approach this one?
In general, there is a "trick" to differentiating a term of the form although I'm not sure it's useful to learn it because it's not that common.
Suppose we had a function with constant. You probably know that the derivative would be .
Suppose we had another function with constant. Again you could differentiate to get .
It turns out that you can calculate the derivative of by first treating constant and differentiating; then treat constant and differentiating; then adding the two results. It's basically like combining the two differentiation rules I just mentioned into one.
In other words:
Applying this to the problem
Usually though you would do such a problem with logarithmic differentiation techniques, and get the solution almost as quickly.
Just in case a picture helps...
Taking logs of both sides of y = ..., we can solve the bottom row for dy/dx in...
... where... (key in spoiler)
Alternatively, as drumist says, and Wolfram also chooses by default, you can use the chain rule for a composite with two inner functions...
... and I fancy this picture makes it palatable...
... is the double version of the ordinary chain rule. As with that, straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed lines similarly but with respect to the (corresponding) dashed balloon expression which is (one of) the inner function(s) of the composite expression. So...
You do, of course, need logarithmic differentiation to see the logic of the double-dashed differentiation, i.e. the logic of...
Don't integrate - balloontegrate!
Balloon Calculus: Standard Integrals, Derivatives and Methods
Balloon Calculus Drawing with LaTeX and Asymptote!
Thank you everybody! A lot of great information in this thread, and it was exactly what I was looking for. Logarithmic differentiation was new to me, but it makes perfect sense now.
I also like to see different approaches to see what feels best. So thank you all for your insights.