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# Finding slope of tangent line with implicit differentiation

Video transcript

We've been doing a lot
of examples where we just take implicit
derivatives, but we haven't been calculating
the actual slope of the tangent line
at a given point. And that's what I want
to do in this video. So what I want to
do is figure out the slope at x is equal to 1. So when x is equal to 1. And as you can imagine,
once we implicitly take the derivative
of this, we're going to have that as
a function of x and y. So it'll be useful
to know what y value we get to when
our x is equal to 1. So let's figure
that out right now. So when x is equal to 1, our
relationship right over here becomes 1 squared,
which is just 1 plus y minus 1 to the
third power is equal to 28. Subtract 1 from both sides. You get y minus 1 to the
third power is equal to 27. It looks like the numbers
work out quite neatly for us. Take the cube root
of both sides. You get y minus 1 is equal to 3. Add 1 to both sides. You get y is equal to 4. So we really want to
figure out the slope at the point 1 comma 1 comma
4, which is right over here. When x is 1, y is 4. So we want to figure out the
slope of the tangent line right over there. So let's start doing some
implicit differentiation. So we're going to
take the derivative of both sides of this
relationship, or this equation, depending on how
you want to view it. And so let's skip down
here past the orange. So the derivative with
respect to x of x squared is going to be 2x. And then the derivative with
respect to x of something to the third power is
going to be 3 times that something squared times
the derivative of that something with respect to x. And so what's the derivative
of this with respect to x? Well the derivative of y with
respect to x is just dy dx. And then the derivative of x
with respect to x is just 1. So we have minus 1. And on the right-hand
side we just get 0. Derivative of a constant
is just equal to 0. And now we need to
solve for dy dx. So we get 2x. And so if we distribute this
business times the dy dx and times the negative 1, when
we multiply it times dy dx, we get-- and actually I'm
going to write it over here-- so we get plus 3 times y
minus x squared times dy dx. And then when we multiply
it times the negative 1, we get negative 3 times y
minus y minus x squared. And then of course, all of
that is going to be equal to 0. Now all we have
to do is take this and put it on the
right-hand side. So we'll subtract it from
both sides of this equation. So on the left-hand side--
and actually all the stuff that's not a dy dx
I'm going to write in green-- so on
the left-hand side we're just left with 3 times
y minus x squared times dy dx, the derivative of
y with respect to x is equal to-- I'm just going to
subtract this from both sides-- is equal to negative
2x plus this. So I could write it as 3 times
y minus x squared minus 2x. So we're adding
this to both sides and we're subtracting
this from both sides. Minus 2x. And then to solve
for dy dx, we've done this multiple
times already. To solve for the derivative
of y with respect to x. The derivative of
y with respect to x is going to be equal to 3 times
y minus x squared minus 2x. All of that over this stuff,
3 times y minus x squared. And we can leave it
just like that for now. So what is the derivative
of y with respect to x? What is the slope of the
tangent line when x is 1 and y is equal to 4? Well we just have to substitute
x is equal to 1 and y equals 4 into this expression. So it's going to be equal
to 3 times 4 minus 1 squared minus 2 times 1. All of that over 3
times 4 minus 1 squared, which is equal to
4 minus 1 is 3. You square it. You get 9. 9 times 3 is 27. You get 27 minus 2
in the numerator, which is going to
be equal to 25. And in the denominator, you
get 3 times 9, which is 27. So the slope is 25/27. So it's almost 1, but not quite. And that's actually what it
looks like on this graph. And actually just
to make sure you know where I got this graph. This was from Wolfram Alpha. I should have told you
that from the beginning. Anyway, hopefully
you enjoyed that.