When to Use the Composite Function/Chain Rule for Derivatives

Ah, the composite function rule/chain rule. (Wikipedia) (Mathematics for Economists).

When would you want to use the composite function/chain rule? (Note: I’m no math expert, so take this all with a grain of salt). Well, if you have a function that’s a function of another function, i.e. a composite function, sometimes the easiest way to find the derivative of the composite function is to use the composite function rule/chain rule.

From Wikipedia:

In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. The functions f: XY and g: YZ can be composed by first applying f to an argument x and then applying g to the result. Thus one obtains a function g o f: XZ defined by (g o f)(x) = g(f(x)) for all x in X. The notation g o f is read as “g circle f“, or “g composed with f“, “g following f“, or just “g of f“.

It may be helfpul to think about what a function is. Functions are generally formulas that you apply to some input, and which “map” the input to some output. The “value” of the output, the dependent variable, is usually named some variable like y, and the name of the function is usually something like f or g; the input, the independent variable, is usually named some variable like x, inside parentheses next to the name of the function, like f(x). From the http://www.math.csusb.edu/ website:

A function (or map) is a rule or correspondence that associates each element of a set X called the domain with a unique element of another set Y called the codomain. We typically give the rule a name such as a letter like f or g (or any letter of your choice) or a name agreed upon by convention like sine or log or square root.

Now, functions can be very simple, such as y=f(x)=x, in which case the function basically doesn’t do anything but map x back to itself. You can have more complicated functions such as y=f(x)=x^3 + 2x + 5, a polynomial, which does quite a few things to the input x before outputting the output value y.

Functions are interesting because basically anything in a mathematical expression can be called a function. Take y=x^3 + 2x + 5 for example. You could say x^3 is a function which maps x to some variable z, and you could name the function g(x). You could say 2x is a function which maps x to some variable u, and you could name the function h(x). You could even say 5 is a function which maps x to the constant 5 each time and name the function i(x) and the name the constant c. You can write 5 as a function of x here if you want to, c=5 * x^0=5*1=5. So pretty much anything in a math expression can be called a function, even constants.

So what about composition of functions? This is another area where I think you can basically find a function to be a composition of functions whenever you want–but there are only certain circumstances in which it matters enough for you to think about using the composite function rule.

One example of a situation in which you have a noticeable composite function is when instead of a lone x or some other independent variable within the parentheses of the function notation, you have other things going on, such as y=f(5x) instead of just y=f(x). In this case, the 5x within the parentheses is a whole other function, you could name the function g for example, and name the output of the function g(x) a dependent variable such as u, and then you would have u=g(x)=5x. Then since y=f(5x), and 5x=u, y is a function of u, a function of the function g(x), and also a function of x, since u is a function of x. In this case we have the composite function y=f(u)=f(g(x))=f(5x).

Now, here’s how to use the composite function rule/chain rule (see Wikipedia and Mathematics for Economists). To find dy/dx, you can first find dy/du then multiply that times du/dx. What if y=f(x)=u^2 and u=g(x)=5x? Then y=f(u)=f(g(x). By the power rule, dy/du would be 2u. Then, where u=g(x)=5x, du/dx would equal 5. By the composite function rule, the derivative dy/dx = dy/du * du/dx = 2u * 5 = 2*5x *5 =50x. This is why I said that there are some cases in which you want to use the composite function rule and in other cases you won’t need to think about it: in this case it might have been simpler to distribute the power of 2 in the beginning, so if we had y= (5x)^2, then y=25x^2, and using the power rule then dy/dx=50x, which is what we got by using the composite function rule above. So sometimes you can simplify first or figure the problem out without explicitly using the composite function/chain rule, and other times it’s easier to start out by using the composite function/chain rule.

When you are multiplying or dividing terms with the variable you are differentiating with respect to, when you are multiplying different functions (see the above about how just about anything can be called a function), in order to differentiate the resulting function, a function which is a product or quotient of two other functions, you can use the product and quotient rules. Once again, you only need to use these rules when it would be easier than multiplying out or dividing out the functions, or when the functions can’t be simplified any further. For example, if you had y=5x^2 * 3x^3 you might as well just multiply this out and then take the derivative of the result. You could have 5x^2 * 3x^3=15x^5, then use the power rule to get dy/dx=75x^4. or you could use the product rule to get the same result, but it would take more effort. You could even use the product rule on y=f(x)=5x, since 5=5*x^0, and here y =5*x^0 * x, in case you were wondering; there are many functions where there’s no point in using the product rule. But if the functions you start with are complicated enough, it can be simpler and easier to use the product rule to begin with instead of multiplying out the functions then taking the derivative of the product. (See product and quotient rules, Wikipedia and Mathematics for Economists). And to sum up, when the output of one function is the input into another function, then you use the composite function rule/chain rule to find the derivative. (See composite function rule/chain rule, Wikipedia and Mathematics for Economists).

Also see: A Closer Look at the Basics of Functions and Derivatives


Some Wayfinding and Survival Strategies for Finding Your Way to a Destination

1. If you have planned a trip, but have moved to a new, unexpected location, check your map to see how your course and plans have been altered, and strategize about how you must now adjust/alter your plans and strategies, update your bearings so that you can successfully find your way to your planned destination. I.e., if you walk past a place you had anticipated walking to, you will then have to adjust your plans, look on a map to find your new bearings. I.e. we walked down past Division. We should have taken a minute to check a map to see how far we were from North. Obviously from the map we would have seen that we were now very, very far from North and should have adjusted our plans accordingly.

2. Discuss with your trip mates that plans have been altered and confer about what the appropriate next strategy is. Do not foolheartedly decide you know exactly where you are going when your plans/location have been altered, without conferring with your trip mates. They may have insight and knowledge you do not. Do NOT under any circumstances fake like you know where you are going or how to get there, with some vague idea such as “we just need to find a large street I am familiar with” when you are in an unfamiliar part of a town, large city, unknown countryside, etc. There are many, many streets and neighborhoods and locations you will not be familiar with and such is a terrible heuristic unless you intentionally want to wander for miles. But your trip mates must consent to such wandering. You must warn them if you are using such a heuristic; they may have better ideas.

3. If you find yourself in a new, unexpected location from a known location, consider backtracking to a known location, especially known safe locations with clear, known connections to where you want to go. I.e. if you were in a location with links to a subway station or highway you know will get you to your destination, consider backtracking to the known location; even if backtracking a little, it may save you a great amount of time and wandering to backtrack. I.e. two wrongs don’t make a right. Unfocused, disorganized, uncalm, unthinking fussing with tangled, knotted yarn may make the yarn only more knotted and tangled. Work slowly, logically, calmly to untangle the yarn. Likewise, if you start wandering from a new position you are unfamiliar with, you may get more and more lost; if you can backtrack to a location with known links to your destination, that may be well worth it. Especially if you see a bus going back to the known location, take it. Do not just plow ahead into unknown, uncharted territory. Never, ever say, “let’s just look for a large street that I know” in an unfamiliar part of town in a large city where you may be in an area with no large streets that you know. It may be simply Hubris and ignorance that make you think you will soon come upon a large street that you know. Do not waste the time and endanger the safety of your trip mates. Clearly communicate exactly what you do and do not know about your current location and means and prospects and strategies for making your way to your desired location.

4. In urban centers in the United States, faced with the choice of walking back to busy city streets with subways stations and bus routes, you probably when trying to find your way to a destination never want to walk along unknown paths along-side freeways, where no one walks, especially when they lead to strange paths under railroads, underpasses, among abandoned side streets and nooks where there are only industrial structures and facilities, abandoned cars, away from civilization. No one ever is in those areas except for drug dealers, gangs, and homeless people. Beware of strange paths alongside freeways where you do not see other people.

5. If you are on an unfocused trip to a destination and have found yourself taken on a strange path alongside some urban freeway, i.e. a no-mans-land, immediately note that you have probably been using terrible way-finding strategies and are officially lost and may be about to be mugged or taken advantage of. If someone has led you down such an abandoned/deserted/industrial/out-of-the-way path you should officially relieve them of any notion that they know where they are going. They are have probably officially taken you on a dangerous trip down a dangerous route, they officially don’t know where they are or where they are going; they don’t know what they are doing or talking about. Beware such people and their faulty logic. They may get you lost and killed. They may be suffering delusions or may be temporarily sick in the head. Find your way back to a safe known location, city center, etc. Ask safe, friendly-looking people for directions. Get on a bus to a known safe location with trains, subways, cabs, safe people. Don’t wander into some abandoned industrial city area where there are bound to be stray dangerous dogs etc.

6. If your guide starts saying things like, “If we just head down this street we will get to xyz” where xyz is where you want to get to, and is miles away, you are officially screwed. Yes, if you are in New York and you head West you will eventually reach the Pacific Ocean. Consider the fact that you should never have listened to such a person in the first place, that their logic may be systematically faulty, and wonder why you ever thought they might know where they are going. Consider that in any trip to a destination you may need to be prepared to find you way to the destination or back to where you started alone, without any guide or companion you started out with. Like the Boy Scouts say, always be prepared. Finding a destination, whether physically or mentally, will always take shrewed and strategic planning and attention. Don’t blindly trust guides who may turn out to be charismatic, who may act like they know what they are doing, but are just the blind leading the blind. Your job is to not be blind.
For wilderness survival there are books such as : Man vs. Wild
Wikipedia: Wayfinding

More Awesome Hip-hop/Dance/Dub/Soul mixes by Klenderfender

Enigmatic DJ/mixmaster/noise manipulator Klenderfender at the Musical Coco Basket blog has been putting out a whole bunch of incredible mixes. Here’s Messy Mix II, lots of great stuff I’ve never heard and some I have heard, taking it back to some great 80s/90s classic/obscure hip hop, dance, dub and reggae, breakbeat, and soul:

D 1 – Crack Bong

Mutabaruka and African Headcharge – What is the Plan? (Version)

Richie Spice – Marijuana (Remix)

Shinehead Real Rock (Irie Ext.)

Tek – Nothings Gonna Change (Inst.)

Grace Jones – Cry Now Laugh Later

Cypress Hill – Real Estate (Inst.)

Prime Minister Pete Nice and Daddy Rich – Rap Prime Minister and Daddy Rich (Inst.)

Simon Harris – Bass (Bomb the House Mix)

28th St. Crew – Inch By Inch

Chill Rob G – The Power (Radio Edit)

Geeneus – Congo

TTC – Catalogue

Congo Natty – Lion In The Jungle Side 4 (White Label)

Wiley – What Do You Call It? (Inst.)

Roots Manuva – Colossol Insight (Jammer Remix + Revox)

Sin – With You (Inst.)

Detroit Emeralds – Baby Let Me Take You

Dobie – The Ride (Inst.)

Freda Payne – Band of Gold

N Tyce – Hush Hush Tip (Inst.)

C. O. D. – Crime Don’t Pay (Inst.)

AMG – Vertical Joyride (Fat Booty Inst.)

Parliament- Flashlight

Joe Sinister – Under the Sun (Inst.)

Barry White – It’s Ecstasy When You Lay Down Next To Me

Pebbles – Girlfriend

Now must go listen to some 808 State last.fm similar artists channel…

The Ruination of Tokion Magazine

I totally agree with most the below review (I liked Tokion up until the format change, mostly).  I used to enjoy reading Tokion magazine–there would be interesting articles on underground musicians, graphic designers, international culture, etc. Sort of like a cross between I.D. magazine, a less crass and crude Vice Magazine, Interview magazine, and Giant Robot. Um, then inexplicably the magazine changed its size to huge and unwieldy (much larger than 8.5 x 11″), and became almost exclusively a fashion magazine, minus interesting articles on music, art, and underground culture. They used to write about some art magazine/underground culture magazine they really liked from the 70s, their new direction seems 100% removed from that type of magazine. Hopefully they regain their senses and previous unique and interesting focus.

From Animal New York

Tokion, the magazine that forgot what it was about—Japanese cool, freaky fashion, and cool art that made it a standout when first published—has a “NEW LOOK” and “NEW FEEL” as the coverlines emphatically states begs. However, after a quick thumb through, you realize quickly that Tokion is still not breaking any NEW ground. Basically, the mag now costs $5.99, is no longer perfect bound, grew to a slightly more awkward size, doesn’t have the editor in chief writing every article, and still maintains the editorial and visual blandness that’s kept it in a perpetual state of irrelevance since investor Larry Rosenblum took the reigns and sent Adam Glickman on his way with a $2 million check a few years ago.

Spike in Vain–best postpunk pre-emo experimental hardcore band ever

Well, in my book Spike in Vain’s Disease in Relative from 1984 is the most-underrated punk/postpunk/experimental/hardcore record ever made! It’s the perfect blend of styles somewhere in the neighborhood of early Minutemen, Black Flag, Joy Division, and Bauhaus, with touches of Killing Joke, King Crimson and mid-period Voivod, Drive Like Jehu, and Sonic Youth. It’s crazy and crazy good. I hope they rerelease it someday. If you’re interested in hearing this amazing record, someone has put up mp3s of it on the megaupload-like mediafire: Spike in Vain–Disease is Relative.
Great news via the Good bad Music for bad, bad Times! blog:

Here’s an email that I received from Robert Griffin who now runs scatrecords.com and was the main man behind Spikes In Vain. Go to his website – there’s a limited supply of original Spike in Vain 12″s available!


“Very cool blog you’ve got there, nice to see some love for that old scene, thanks. I’ve gone on to play lots of other music, but the attitudes and aesthetic of those days will always guide me.

Sometime next year I’m going to release all the Spike in Vain recordings on cd. After Disease Is Relative, there was a cassette-only album, “Jesus Was Born in a Mobile Home,” and there is also an unreleased third album, “Death Drives a Cadillac.” There may be a DVD in there also, I recently got a video of an almost complete 1983 gig. There are some other odds and ends too, it will most likely be a 3 disc set. If there’s enough interest I might press up some of the unreleased material on vinyl also.

Some other info for you – Outerwear were kind of a sister band to SiV. I played drums, Chris Marec on gtr/vox, and Beth Scarf on bass. There was a cassette album called “The Outerwear Limits.” Our slogan “loud slow bad rules” heh.. We had a great time pissing off all the “correct” punks. But Beth and I did eventually learn our instruments, so we started doing things like wrapping ourselves in bandages so that we couldn’t play so well. Very fun band and Chris wrote some pretty cool songs for it.

Funny, the Guns were the youngest band on that compilation, 14 or 15 years old at most, and both the main guys have passed on. Dave died of a brain aneurism in his 20s, and Scott died last fall of a heart attack. Just before he died he put together a Guns cd – all their original recordings, maybe a dozen tunes, then re-recorded versions from a reunion a few years ago. Not sure how you’d get one of those. These two were also the drummer & bassist in The Dark. Small scene, lots of inbreeding!

There’s also a 2CD by The Dark on Grand Theft Audio called Scream Until We Die.



Posted on 20-May-08 at 11:40 am

Spike in Vain and the Dark are two of my favorite bands ever…totally like the mighty DRIVE LIKE JEHU but circa 1984! Totally predated a lot of the whole DC emo/Fugazi/Rites of Spring stuff. Better than 99% of indie postpunk stuff out there, kind of undiscovered in the same way great Live Skull albums are. Robert Griffin is one heck of a experimental postpunk post-prog hardcore noise guitarist! I like their stuff as much as the mighty Live Skull, will appeal greatly to fans of bands like Drive Like Jehu, Sonic Youth, the Gordons, early Killing Joke, etc. .

The Best Hip Hop Samples Ever

Kate Bush–Blow Away (for Bill)

Check out the pianos at the beginning and end of the song, ’nuff said.  An amazing song all around.

Culture Club–Victims. Check out the intro pianos, the drums and epic vocals and piano at 2:17, the amazing part with the piano and the strings starting at 3:48, and then the part at 4:19…

I think Just Blaze should make a beat out of this and get Jay-Z on it.

Oh, there’s a version with no Boy George vocals, an instrumental (not sure if it’s exactly the same), by Culture Club–Romance Revisited. Well, they put too much oboe or something over the intro piano in the Romance Revisited version, and they took out the awesome drums and stuff and the crazy female vocals. Stick with the original version with the drums and vocals and everything.

Al Stewart – The Year of the Cat

Check out the amazing piano at the beginning of this song, just ready to be sampled for some hip hop song. Great epic song all around, nice guitar AND sax solos. The vocals sound almost like some Manchester music in the 90s like Oasis or Blur, or some new wave band, but better, more classic classic rock, more John Lennon mixed with Donovan or Cat Stevens or something.

Ides of March — Vehicle

Check out the crazy horns at the beginning. This would sound something like Jay-Z’s “Roc Boyz” is done properly. Great guitar solo too.

Also see: Francis Lai-L’aventure soundtrack

And the opening track on George Hirota’s Sahasurara album

Also see the bongos at the beginning of the T-rex song Rip Off

John McLaughlin–the chorus,piano part etc, in Desire and the Comforter

Via DJ Letters and Science

Yes — Close to the Edge Pt. 1, Pt. 2

Yes–Close to the Edge Pt.1

Yes–Close to the Edge Pt.2

And then God told Yes to create Close to the Edge Pt. 1…This is hand-down one of the best pieces of experimental music, classic rock, prog rock, indie, indie-metal, prog-metal, postpunk ever made…(of course, having been made in 1972, this song predates most of those musical genres).   All of the music of bands like Voivod, Don Caballero, Sonic Youth, Drive Like Jehu, the Minutemen, etc., could probably fit in this one song. I’ve always thought Mike Watt must have been a huge Yes fan in the Minutemen, and that the Minutemen were fantastic at making one-minute-long prog rock opuses (so many parts and ideas fit into so little time!!!). It’s weird that I never really heard people refer to early Minutemen as prog rock, but that’s what it so obviously is (at least up until Double nickels on the Dime?). Yes–Closer to the Edge, for fans of the above-mentioned bands and obviously bands like Gentle Giant, King Crimson, Magma, etc.