Does this differentiation trick work?

[Archosaur]

Hey all,

I came up with a trick for fast differentiation, and it's worked every time I've used it, but I was hoping you guys could possibly formally prove that it works, or at least show me a case where it doesn't?

It goes like this:
(Sorry I can't figure out Latex)

When you have an expression of the form
f(x)=a^bc^de^f...y^z

then f`(x)=f(x)*(b(a`/a)+d(c`/c)+f(e`/e)+...z(y`/y))

It rarely gives the derivative in it's simplest form, but it allows me to tear quickly and mechanically through the problem.

Does anyone see any issues with this?

Also, has anyone seen anything like this? I'm not far along in math, so I would be astonished if I came up with something new, but I have taken up to calc 3 and no teacher has ever taught this.

 

[arildno]

It is a subtype of the general logarithmic differentiation trick:

Let
$$y=\ln(f(x))$$

Therefore, we have:

f'=f*y', which is readily established.

Your expression is correct, as long as your original exponents are constants.

To take another example with the logarithmic differentiation tric, look at the following:
$$f(x)=x^{x}\to{y}(x)=x\ln(x)\to\frac{dy}{dx}=\ln(x)+1\to\frac{df}{dx}=x^{x}(\ln(x)+1)$$

which is the correct answer.

This can also be gained by another wacky, yet valid differentiation trick:

We first differentiate with respect to the "x" in the base, treating the "x" in the exponent as constant, then vice versa, and finally add the results together:
$$\frac{df}{dx}=x*x^{x-1}+x^{x}\ln(x)=x^{x}(\ln(x)+1)$$

That trick utilizes chain rule for partial differentiation.

 

[Borek]

Compare Feynman's tips on physics, chapter 1-4.

Now - calculate probability that after many years of thinking about it I have bought complete Feynman's lectures, that I started with browsing Feynman's tips book and not other one, and that I found this chapter about 24 hours after I have seen post about your trick

 

[Archosaur]

Compare Feynman's tips on physics, chapter 1-4.

Now - calculate probability that after many years of thinking about it I have bought complete Feynman's lectures, that I started with browsing Feynman's tips book and not other one, and that I found this chapter about 24 hours after I have seen post about your trick

That's pretty remarkable! I was going to go about estimating the chances with the knowledge that 1.5 million copies of his lectures have sold since they came out, but then I found that his "Tips on Physics" can also be bought separately, and I can't find sales data on it. I can guess, though, that the chances are: "not very good"

But, hey, to throw another coincidence on the pile, I'm currently saving up for his lectures! I haven't eaten out for about a month.

 

[arildno]

No thanks to me for showing why the trick works!

 

[Borek]

No thanks to me for showing why the trick works!

Arild, you are the best

 

[arildno]

sniff, thanks, Borek!

Hmm, wait a sec:
Is that just the newbie mentor trying to ingratiate himself?
ponder, ponder, mumble..probably not..

 

[Archosaur]

No thanks to me for showing why the trick works!

Ahh! How rude of me! Thanks, seriously. I totally see where it comes from, now.

I would throw you a "Thank you / I'm sorry" party, but coordinating it would be a logistical nightmare, so the best I can do is offer the following array of smilies:


I hope you find it to be an acceptable party-substitute.

 

[arildno]

Oh, it is a most acceptable party substitute, bow&thanks!