- [Voiceover] Let's say that Y is equal to
log base four of X squared plus X.
What is the derivative of Y
with respect to X going to be equal to?
Now you might recognize immediately that
this is a composite function.
We're taking the log base four, not just of X,
but we're taking that of another expression
that involves X.
So we could say
we could say this thing in blue
that's U of X.
Let me do that in blue.
So this thing in blue
that is U of X.
U of X is equal to X squared
plus X.
And it's gonna be useful later on to know
what U prime of X is.
So that's gonna be
I'm just gonna use the power rule here
so two X plus one
I brought that two out front
and decremented the exponent.
Derivative with respect to X of X is one.
And we can say the
log base four of this stuff
well we could call that a function V.
We can say V of
well if we said V of X
this would be log base four
of X.
And then we've shown in other videos
that V prime of X
is, we're gonna be very similar that if this
was log base E, or natural log,
except we're going to scale it.
So it's going to be
one over
one over log base four.
Sorry, one over
the natural log.
The natural log of four
times X.
If this was V of X, if V of X was just
natural log of X, our derivative would be one over X.
But since it's log base four
and this comes straight out of the change of base
formulas that you might have seen.
And we have a video where we show this.
But we just scale it in the denominator
with this natural log of four.
You think of scaling the whole expression
by one over the natural log of four.
But we can now use this information because Y
this Y can be viewed as
V of
V of.
Remember, V is the log base four of something.
But it's not V of X.
We don't have just an X here.
We have the whole expression
that defines U of X.
We have U of X right there.
And let me draw a little line here
so that we don't get those two sides confused.
And so we know from the chain rule
the derivative Y with respect to X.
This is going to be
this is going to be the derivative of V
with respect to U.
Or we could call that V prime.
V prime of U of X.
Let me do the U of X in blue.
V prime of U of X
times U prime of X.
Well, what is V prime of U of X?
We know what V prime of X is.
If we want to know what V prime of U of X
we would just replace wherever we see an X
with a U of X.
So, this is going to be equal to
V prime of U X, U of X.
And you just do is you take the derivative
of the green function with respect to the blue function.
So it's going to be one over
the natural log of four.
The natural log of four.
Times, instead of putting an X there
it would be times U of X.
Times U of X.
And of course, that whole thing
times U prime of X.
And so, and I'm doing more steps just hopefully
so it's clearer what I'm doing here.
So this is one over
the natural log of four.
U of X is X squared plus X.
So
X squared plus X.
And then we're gonna multiply that
times U prime of X.
So times two X plus one.
And so we can just rewrite this as
two X plus one over
over
the natural log of four.
The natural log of four
times X squared plus X.
Times X squared
plus X.
And we're done, and we could distribute
this natural log of four
if we found that interesting.
But, we have just found the derivative of Y
with respect to X.