Hi everyone!
Today we're going to talk about how to calculate
the second derivative of a parametric curve.
To complete this problem, we'll find the first
derivative of the parametric curve and then
use the first derivative to find the second
derivative.
Let's take a look.
So, in this particular problem, we've been
given the two equations, y equals two t minus
one and x equals t squared plus t to define
our parametric curve.
We need to find the second derivative of the
parametric curve but in order to do that of
course we'll have to find the first derivative
first.
Remember that in order to find the first derivative
of the parametric curve, which we'll call
dy over dx, we need to find the derivative
of the y equation with respect to t and we
need to find the derivative of the x equation
with respect to t and to find the derivative
of the parametric curve in general we'll divide
the derivative of the y equation by the derivative
of the x equation.
So both these derivatives are fairly basic.
We can see that the derivative of the y equation
with respect to t is just two.
If we take the derivative of two t minus one
term by term, we get two minus zero which
of course just simplifies to two so our numerator
will be two.
Our denominator will be the derivative of
the x equation with respect to t.
So when we take the derivative of t squared
plus t we get two t plus one.
Now that we found the first derivative of
this parametric equation, we can go ahead
and find the second derivative of the parametric
equation.
We'll denote it with d squared y over dx squared
and we'll say that that is equal to d/dt of
dy/dx over dx/dt.
Now, all this formula tells us is that the
numerator of our second derivative will be
the derivative of our first derivative so
we'll go ahead and take the derivative of
two over two t plus one.
And our denominator will just be dx/dt which
we've already calculated up here, dx/dt, we
know that that is equal to two t plus one.
So, this shouldn't be too difficult.
Our numerator, again, will be the derivative
of two over two t plus one which we'll use
the quotient rule to find.
Remember that the quotient rule tells us to
take the derivative of our numerator so the
derivative of two which is just zero, and
multiply that by the denominator so two t
plus one, we'll subtract from that the numerator,
two, times the derivative of the denominator,
and the derivative of two t plus one is just
two, and then we divide all of that by the
square of the denominator, so two t plus one
squared.
That is our numerator of our second derivative.
The denominator is just dx/dt which we've
already calculated as two t plus one.
So now we can go ahead and simplify this.
As you can see, zero times two t plus one
of course will go away, we'll get zero there.
So we'll just get negative four, on top here,
divided by two t plus one squared.
And we have this two t plus one in our denominator,
dividing by two t plus one is the same of
course as multiplying by one over two t plus
one.
So we can see here that well end up with negative
four in our numerator divided by the quantity
two t plus one cubed in the denominator.
And that is it, that's our second derivative.
So I hope you found that video helpful.
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