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## Chain rule

# Worked example: Derivative of ln(√x) using the chain rule

AP.CALC:

FUN‑3 (EU)

, FUN‑3.C (LO)

, FUN‑3.C.1 (EK)

## Video transcript

- [Voiceover] So we have here f of x being equal to the natural
log of the square root of x. And what we wanna do in this video is find the derivative of f. And the key here is to
recognize that f can actually be viewed as a composition
of two functions. And we can diagram that
out, what's going on here? Well if you input an
x into our function f, what's the first thing that you do? Well, you take the square root of it. So if we start off with
some x, you input it, the first thing that you do,
you take the square root of it. You are going to take the
square root of the input to produce the square root of x, and then what do you do? You take the square root and then you take the natural log of that. So then you take the natural log of that, so you could view that as inputting it into another function
that takes the natural log of whatever is inputted in. I'm making these little squares to show what you do with the input. And then what do you produce? Well you produce the natural
log of the square root of x. Natural log of the square root of x. Which is equal to f of x. So you could view f of
x as this entire set, or this entire, I guess you could say, this combination of
functions right over there. That is f of x, which is essentially, a composition of two functions. You're inputting into one function then taking that output and
inputting it into another. So you could have a function u here, which takes the square root
of whatever its input is, so u of x is equal to
the square root of x. And then you take that output, and input it into another
function that we could call v, and what does v do? Well it take the natural log
of whatever the input is. In this case, in the case
of f, or in the case of how I just diagrammed it, v
is taking the natural log, the input happens to be square root of x, so it outputs the natural
log of the square root of x. If we wanted to write
v with x as an input, we would just say well
that's the natural log, that is just the natural log of x. And as you can see here, f of x, and I color-coded ahead of time, is equal to, f of x is equal to, the natural log of the square root of x. So that is v of the square
root of x, or v of u of x. So it is a composition
which tells you that, okay, if I'm trying to
find the derivative here, the chain rule is going to be
very, very, very, very useful. And the chain rule tells
us that f prime of x is going to be equal to the derivative of, you can view it as the outside function, with respect to this inside function, so it's going to be v prime of u of x, v prime of u of x, times the derivative
of this inside function with respect to x. So that's just u prime, u prime of x. So how do we evaluate these things? Well, we know how to take
the derivative of u of x and v of x, u prime of x
here, is going to be equal to, well remember, square root
of x is just the same thing as x to 1/2 power, so we
can use the power rule, bring the 1/2 out from so
it becomes 1/2 x to the, and then take off one
out of that exponent, so that's 1/2 minus one
is negative 1/2 power. And what is v of x, sorry,
what is v prime of x? Well the derivative of
the natural log of x is one over x, we show
that in other videos. And so we now know what u prime of x is, we know what v prime of x is,
but what is v prime of u of x? Well v prime of u of x,
wherever we see the x, we replace it, let me write
that a little bit neater, we replace that with a u
of x, so v prime of u of x is going to be equal to, is going to be equal to one over u of x, one over u of x, which is equal to, which is equal to one over, u of x is just the square root of x. One over the square root of x. This thing right over
here, we have figured out, is one over the square root of x, and this thing, u prime
of x, we figured out, is 1/2 times x to the negative 1/2, and x to the negative 1/2,
I could rewrite that as 1/2 times one over x to the
1/2, which is the same thing as 1/2 times one over
the square root of x, or I could write that as one
over 2 square roots of x. So what is this thing going to be? Well this is going to
be equal to, in green, v prime of u of x is one
over the square root of x, times, times, u prime of
x is one over two times the square root of x, now what
is this going to be equal to? Well, this is going to be equal to, this is just algebra at this point, one over, we have our
two and square root of x times square root of x is just x. So it just simplifies to one over two x. So hopefully this made sense, and I intentionally diagrammed it out so that you start to get that
muscle in your brain going of recognizing the composite functions, and then making a little bit more sense of some of these expressions
of the chain rule that you might see in your calculus class, or in your calculus textbook. But as you get more practice,
you'll be able to do it, essentially, without having
to write out all of this. You'll say okay, look,
I have a composition. This is the natural log
of the square root of x, this is v of u of x. So what I wanna do is I
wanna take the derivative of this outside function with respect to this inside function. So the derivative of
natural log of something, with respect to that something,
is one over that something. So it is one over that something, the derivative natural log of something with respect to that something
is one over that something, so that's what we just did here. One way to think about it,
what would natural log of x be? Well that'd be one over x,
but it's not natural log of x. It's one over square root of x, so it's going to be one
over the square root of x, so you take the derivative
of the outside function with respect to the inside one, and then you multiply that
times just the derivative of the inside function with respect to x. And we are done.