Derivative of triple composition | Taking derivatives | Differential Calculus | Khan Academy
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Derivative of triple composition | Taking derivatives | Differential Calculus | Khan Academy

October 12, 2019


Consider the functions f and
g with the graphs shown below. If capital G of x is equal to g
of f of x whole thing squared, what is the value of G
prime, capital G prime, of 5? And I encourage you to
now pause this video and try to solve it on your own. So let’s try to think
through this somewhat complicated-looking function
definition right over here. So we have capital G of x. And actually, let me
write it this way. Let me write it this way,
I’ll do it in yellow. We have capital G of x is
equal to this quantity squared. What we’re squaring
is g of f of x. g of f of x is what
we’re squaring. Or another way to
write G of x, If h of x were to be equal
to x squared, we could write G of x is equal
to h of this business, h of g of f of x. Let me just copy
and paste that so I don’t have to keep
switching colors. So copy and paste, there we go. So this is another
way of writing G of x, where
whatever g of f of x, we input then to h of x, which
is really just squaring it. So there’s a couple of
ways that we can write out the derivative of capital
G with respect to x. And you could
imagine this is going to involve the chain rule. But I like to write it
out, just to clarify in my head what’s going
on and to make sure that it actually
makes some sense. So one thing that
we could write, we could write the
derivative of G with respect– I’ll mix
notations a little bit– but I’ll write the derivative
of G of x with respect to x is equal to the
derivative of this whole thing. So let me copy and paste
it, copy and paste. It’s equal to this derivative
of this whole thing with respect to what’s inside of
that whole thing. So if you wanted to treat
g of f of x as a variable, so with respect to that. So copy and paste. So it’s going to be the
derivative of this whole thing with respect to g of f of
x times the derivative of g of f of x with respect to f
of x, with respect to– I’ll just copy and paste this
part, whoops– with respect to f of x. And I like to write this out. It feels good. It looks like these are rational
expressions with differentials. It’s really a notation more
than to be taken literally. But it feels good, or
at least in my mind it’s a little bit more intuitive
why all of this works out. So with respect to f of x times
the derivative of– and I’m using non-standard
notation here, but it helps me really
conceptualize this– times the derivative of f of
x with respect to x. Or another way we
could write this is G prime of x is
equal to h prime of g of f of x, h
prime of– actually, let me do it here–
h prime of this. So copy and paste,
h prime of that, times g prime of f of x,
times g prime of this. So copy and then paste. So times g prime of that. Put some parentheses there. Times f prime of x. And I like writing it this
way, because you notice if these were– and once
again, this is more notation, but it gives a sense
of what’s going on. If you did view
these as fractions, that would cancel with that. That would cancel with that. You’re taking the derivative
of everything with respect to x, which is exactly
what you wanted to do. And let me put some
parentheses here so it makes a little bit
clearer what’s going on. But this thing, in
my brain, I like to translate that
as, well, that’s just h prime of g of f of x. This is g prime of f of x. This is f prime of x. And going from this
to try to answer your question, the question
that they’re asking us actually isn’t too bad. So we want to know,
what’s G prime of 5? So everywhere we see an
x, let’s change it to a 5. So we’re going to say, we need
to figure out what G prime of 5 is. G prime of 5 is equal
to– and actually, let me just copy and
paste this whole thing. So copy and paste. And so let me, everywhere
where I see an x, I’m going to
replace it with a 5. So let me get rid of that. Let me get rid of that. And let me get rid of that. And so I have a 5, a 5, and a 5. So what is f of 5? f of 5 is equal to negative 1. So this right over here
simplifies to negative 1. This right over here
simplifies to negative 1. And what’s f prime
of negative 5? Well, that’s the slope
of the tangent line at this point right over here. And we see that the
derivative, or the slope, of the tangent line here is 0. So this right over here
is going to be equal to 0. Now that’s really interesting. So we could keep trying to,
well, what’s g of negative 1? What’s g prime of negative 1? You could see g of
negative 1, g of negative 1 we see is negative 1. g prime
of negative 1 is the slope here, which is also negative 1. Then we could calculate
h prime of these values, et cetera, et cetera. But we don’t even
have to do that. Because this is the
product of three things, and one of these things
right over here is a 0. So 0 times anything
times anything is going to be equal to 0. Another way of thinking
about it is, f of x is isn’t changing when
x is equal to 5. If f of x isn’t changing
when x is equal to 5, then the input into the g
isn’t going to be changing. So the g function isn’t going
to be– in the composition g of f of x– isn’t
going to be changing. And so h of g of f of x
isn’t going to be changing. So g of x isn’t
going to be changing. And so the derivative of
capital G of x at x equals 5 is going to be equal to 0.

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