How To Understand Derivatives: The Product, Power & Chain Rules

Kalid, you are the embodiment of enlightenment. You have been such an inspiration on my journey through math and I cannot thank you enough. Please keep doing what you’re doing!

That said, I speak as someone who passed a college level Calc course but never understood a lick of anything I was doing (dunno how that’s possible). Simply memorizing everything, I just remember an overwhelming urge to bash my brains out on my school desk. Actually getting that same urge now.

Crazy ehh?

We know f’(x) = dy/dx = 2 * x
At x = 10 the “output wiggle per input wiggle” is = 2 * 10 = 20. The output
moves 20 units for every unit of input movement.
If dx = 0.1, then dy = 20 * dx = 20 * .1 = 2

Umm what?
If F(x) = x^2 and x=10 then the result of that would be 100, not 20.
If 2*10=2 then the output would move 2 units for every unit of input movement, not 10, this doesn’t make any sense at all.

@andy: The payoff is understanding something you didn’t before! :slight_smile: Yep, if the analogy works for you, it works.

Thank you! I really wanted to understand how and why these rules work, not just how to apply them :smiley:

I finally understand the product rule! That was amazing.

@Alex: Awesome! I’m the same way, the product rule never clicked until it was visualized as area.

To use calculus on any changing system, is it mandatory, that the system “MUST” follow a particular rule of change.
For example, when some system is changing by ratio of 1:2, then one can find out the change as 2:4 or 100:200 finally. What is the rule of change in calculus ? If not, will calculus be able to find an accurate answer every time ?

@Matthias: Glad it helped.

@JJ: Awesome, glad you’re getting a head start! You got it, the math isn’t much more than algebra, it’s just seeing how to put the variables together.

@Vishwas: Calculus is made for instantaneous rates of change, i.e. the rate of change at a certain moment in time. As you move away from that moment, the rate of change varies and is no longer accurate [and you use integration to add-up these constantly-shifting moments].

Still does not make any sense unfortunately - including the mechanical computer video and wiggles. Perhaps it’s hopeless and I will never understand calculus despite wanting to. I was lost earlier. If f(x) = x^2 and input is 10. Wiggle of 0.1 gives wiggle of 2.01 and wiggle of 0.01 gives 0.2001. How do these relate to the derivative?

Hi, am working through the tutorials from naught, to gather an understanding of calculus and within 4 weeks when my course will start. Is there anyone who might help me excel more than is possible on my own via skype or phone? Based in Melb, 24.01.14.

Hi Silrak, there’s a full series on calculus here: http://betterexplained.com/calculus/ which might help.

why you neglect df * dg area?
if rate of change is more then error produces in the way of intuition,

i understand 95% about product rule except this part

what a intuition to produt rule! i really like it except neglecting dy * dg

[…] Can we see a giant function as being parameterized by smaller ones? (See the chain rule.) […]

@raju
Great question! This question, and a few others very much like it, gave me a bit of trouble until I performed a bit of mental ju jitsu to convince myself I understood it. I’ll have a crack at answering this, if it helps great, if not feel free to ignore me, I won’t be offended.
I can answer it with a bit of simple, but creative, algebra. To do so I’m going to need to explain two different ‘sizing’ operations: the integral ( ∫ ) and the differential element (d). I don’t want to throw a bunch of integration at you in trying to explain derivatives, but if you blur out the strict definitions and just look at the ideas, then ∫ and d are just two different ideas that are a kind of compliment to each other.

∫ - add up a whole bunch of things
d - take a thing and cut it into a bunch of ‘itty bitty pieces’, or i.b.p.

Taking a differential element of a pizza, or d(pizza), is just shaving off a little bit, it’s lifting a pepperoni and licking the bottom then replacing it while nobody’s looking. The small amount of pepperoni grease that’s missing is so small compared to the whole size of the pizza that no one notices it’s missing.

In that vein I’m going to talk about taking some i.b.p.'s of a few things. Start with:
h(x) = f(x) * g(x)
h is the full size of h, the output
dh is an i.b.p. of the output

x is the full size of an input
dx is an i.b.p. of an input

I’m going to take a few liberties and do some ‘algebra magic’ with quantities like dx, please understand that what I’m about to insinuate is not technically allowed, and if we followed the ‘mathematically correct’ path we would perform a whole lot of weird calculus operations just to come up with the same result, see Kalid’s ‘Aside’ note about the Engineers nodding and the mathematician frowning.

I’m going to assume you followed Kalid’s logic and got to this part:
dh = fdg + gdf

dh is just an i.b.p. of h, it is the little bit of rectangle you add on to the full size rectangle of f*g.

fdg is the vertical sliver rectangle
g
dh is the horizontal sliver rectangle

So it seems your question becomes the following: if I want to describe all of that i.b.p. of h that I’m adding I can see I need to add the two slivers, but don’t I also need to add that little square? Sure it’s teeny, but the slivers are teeny too, aren’t they?

How about we don’t disregard it, how about we add it in then see why it vanishes and the slivers are big enough to stay.

We should have:
dh = fdg + gdf
but logic tells us we have:
dh = fdg + gdf + (df*dg) that pesky little square

dh = fdg + gdf + (df*dg) **now divide both sides by dx
dh/dx = f * dg/dx + g df/dx + (dfdg)/dx **re-arrange () in 3rd term
dh/dx = f * dg/dx + g *df/dx + df * (dg/dx) **put that 3rd term right after the 2nd
= f * dg/dx + df * (dg/dx) + g * df/dx **factor out the dg/dx
= (f + df) (dg/dx) + g * df/dx **almost there, compare it to what we should have:

       = (f        ) (dg/dx) + g * df/dx

What happens to (f + df) as df gets ‘eensy weensy’? It gets really close to f. The full size of f on its own is indistinguishable from the full size of f plus an i.b.p. of f. No one sees the little bit of pepperoni grease missing compared to the full size of the pie (I live in New England, we call pizza ‘pie’ up here, we also take 'r’s out of words and stick 'em other places they don’t belong: pahk the cah, Delter airlines).

With that pesky little square we have:
dh/dx = (f + df) (dg/dx) + g * df/dx

Without we have:
dh/dx = (f ) (dg/dx) + g * df/dx

But these two are the same!

If I haven’t confused you yet maybe this will throw you off guard (I jest, I really do wan’t you to understand)!
Here’s another reason that little square ( df * dg ) vanishes but the slivers remain. Let’s use an analogy that all of us understand so well on an intuitive level: Boolean Algebra! (Bang head against wall now)

  • means OR
  • means AND
    P(a) means probability event a happens
    P(a) + P(b) means probability of a or b happening
    P(a) * P(b) means probability of a happening AND THEN b happening
    That little square is df * dg, its kind of like take a little chunk of f, then a little chunk of g. It’s like licking the pepperoni, AND THEN a little flea jumps on your tongue and licks a little grease from your tongue. You don’t notice it compared the ‘large’ little bit of grease you got, which itself is small compared to the pie. Just df on its own is you licking the pepperoni. Just dg on its own is the flea licking your tongue. The sum df + dg is either you licking the pepperoni or the flea licking your tongue. But df *dg is licking the pepperoni AND THEN getting licked by the flea, it means comparing the flea’s very little bit of grease to the whole pie.

Hope this helps, or at least that you got the chance to laugh at Calculus for a little while.

Excelsior,
Eric V

Sorry if the formatting is hard to follow. Can’t use tab so I used a bunch of spaces instead to keep my = lined up under each other. The Posting Gnome ate my spaces. He also messed up my comment markers ** at the end of lines.

AAAHHH LIGHTBULB!!! :smiley:

This video helped me get the Chain Rule by working out d/dx(sin(x^2)) compared to d/dx(sin(x)) in a really visual and intuitive way. http://youtu.be/bcGOZLL1v4Y

This is definitely written for people who have already taken calculus and not understood it, versus someone with almost no exposure who is trying to learn. I don’t even think reading over this would be helpful. You assume I already know all this stuff! Do you know of any good place to START learning calculus? Maybe I can come back to this site after I memorize all those rules, if I’m still confused.

Hi Bonnie, yep, this lesson is definitely geared for someone in the tail end of a calculus class. If you’re just starting out, check out:

http://betterexplained.com/calculus/lesson-1

Hope that helps!