today we're going to be talking about how
to find the derivative of a parametric curve
and in this particular example we've been
given the parametric curve defined by these
two functions x equals t sine of t and y equals
t squared plus t so whenever we're finding
the derivative of a parametric curve we need
to remember that the derivative we're looking
for is dy over dx which means take the derivative
of y with respect to x well in this case we
have two different functions normally when
we find dy dx we just take the derivative
of our function because our function is a
function of y in terms of x but in this case
we have two functions one for x and one for
y and they're both in terms of t so the way
that we take the derivative is that we actually
take the derivative of y with respect to t
and we divide that by the derivative of x
with respect to t so what this tells us is
that we'll take the derivative of our function
for y with respect to the variable t we'll
put that here in our numerator and then we'll
take the derivative of the function x with
respect to t and put that in the denominator
so we'll find the derivatives completely separately
but put them into one fraction so the derivative
of our function y with respect to t we're
just using power rule to take the derivative
of t squared plus t so the derivative will
be two t plus one that's an easy one the derivative
of the denominator here which is dx over dt
in other words the derivative of the function
x with respect to t we're going to need to
use product rule because we have two functions
multiplied together or the product of two
functions one function is t and the other
is sine of t so remember that product rule
tells us that when we have two functions that
are multiplied together we have to take the
derivative of the first one multiplying it
by the second then leave the first one alone
and take the derivative of the second and
we take the second and we take the sum of
those two products so we'll take the derivative
of just t by itself and the derivative of
t is one and then we'll multiply that by sine
of t so sine of t and now we'll add to that
the opposite case where this time we take
the derivative of sine of t and we leave t
alone so we'll say times we have t times the
derivative of sine of t which we know to be
cosine of t so we've essentially taken the
derivative and now it's just a matter of simplifying
as much as we can we'll the simplification
on this one is pretty easy what we find is
that dy over dx the derivative of this parametric
curve is just two t plus one divided by sine
of t plus t cosine of t and that's it that's
our final answer so I hope you found that
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