today we're going to be talking about how
to find the derivative of an inverse trigonometric
function in this particular case we're going
to be looking at the function y equals the
inverse sine or arcsine of two x plus one
and as a reminder we're going to be using
the formula for the derivative of arcsine
which tells us that the derivative of arcsine
is equal to one divided by the square root
of one minus x squared now the important thing
to note about this formula and in particular
all of the formulas for the derivatives of
inverse trigonometric functions is that whenever
we have this x inside of inverse trigonometric
function that is tied specifically to the
x here in our result and what that means is
that whatever we have here for our x in this
case we have two x plus one that two x plus
one is going to go in here for x so our result
is not just going to be one divided by the
square root of one minus x squared it's going
to be one divided by the square root of one
minus the quantity two x plus one squared
so whatever we have inside our arcsine function
or our cosine our tangent etc we want to make
sure we plug in for x into the right hand
side here of our formula so the derivative
y prime we'll need to calculate using chain
rule and for the chain rule part we'll need
to recognize arcsine as the outside function
and two x plus one as the inside function
so essentially we'll take the derivative of
the outside function first leaving the inside
function completely untouched and multiply
by the derivative of the inside function so
we know that the derivative of arcsine is
this right hand side over here so what we'll
do is say one over the square root of one
minus and remember here we have to plug in
whatever is on the inside of our arcsine function
here so in this case two x plus one squared
so that is the derivative of the outside piece
it's the derivative of arcsine but now because
we have something more involved inside of
our arcsine function then simply x we don't
have just arcsine of x we have arcsine of
a function here we need to multiply by the
derivative of the inside function so the derivative
of two x plus one is two so we go ahead and
multiply our result here by two and that's
how we apply chain rule to find the derivative
so remember that whatever you have inside
here you're plugging in for x in your formula
and then if you have anything special in here
make sure you multiply but he derivative of
the this inside function and now it's just
a matter of simplification so what we'll do
is we'll simplify what we have under the square
root so we'll get the square root of one minus
keep in mind here that this is two x plus
one times two x plus one and we're just going
to multiply out so we'll get here four x squared
plus two x plus two x is plus four x plus
one and that's all under the square root and
then of course we still have this multiplied
by two so we'll continue simplifying let's
bring the two into our numerator and then
here in our square root we'll have one minus
four x squared minus four x minus one when
we distribute that negative sign now we can
see that we're going to get one and negative
one here to cancel and we'll be left with
y prime is equal to two divided by the square
root of negative four x squared minus four
x what we can do is we can pull out a four
inside of our square root sign so when we
do that we factor out a four we'll get four
times negative x squared minus x and close
that square root what we can do now is take
the square root of the four separately when
we do we'll get two so we'll get two divided
by two times the square root of what's remaining
which is negative x squared minus x and at
this point we can go ahead and cancel the
two from the numerator and denominator that
means that our final answer is y prime is
equal to one divided by the square root of
negative x squared minus x and that's how
we use the formula for the derivative of arcsine
to find the derivative of this inverse trigonometric
function so I hope you found that video helpful
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