We want to find the derivative
of f of x equals inverse
hyperbolic cosecant x
raised to the second power.
So the first thing we should recognize
is that f of x is a composite function,
so we'll have to apply the chain rule
in order to find our derivative.
And when applying the chain rule,
it's important that we recognize
which function is the inner function
and which function is the outer function.
For our function, the inner function
is the inverse hyperbolic
cosecant function here,
and the outer function
is the squaring function,
which means in order
to find our derivative,
we'll have to apply the
general power rule given here.
Notice how this general power
rule includes the chain rule,
where the inner function is equal to u,
and therefore the derivative of u to the n
with respect to x has two parts,
where the first part is
equal to the derivative
of the outer function.
You evaluate it at the inner function.
In the second part,
u prime is equal to the
derivative of the inner function.
So again, go back to our function.
Since this is the inner function,
we'll let this be equal to u.
So if u is equal to inverse
hyperbolic cosecant of x,
then we can think of our
function as just u to the second,
and then apply our general power rule.
But notice how we'll also
have to find u prime,
so let's go ahead and find that now.
To find u prime,
we'll have to use this
derivative formula here.
And notice how this derivative formula
does include the chain rule,
but in this case, u would be equal to x,
and therefore u prime
would be equal to one,
so our function doesn't
require the chain rule.
So u prime would be equal to negative one
divided by the absolute value of x
times the square root
of one plus x squared.
So our derivative, f prime of x,
is equal to the derivative of u squared
with respect to x, so
we're going to multiply
by the exponent, that would be two
times u to the first.
But u is equal to inverse
hyperbolic cosecant of x
times u prime, which we found here.
Negative one divided by
the absolute value of x
times the square root
of one plus x squared.
So again, notice how we've
applied the chain rule,
where this part is the
derivative of the outer function
evaluated at the inner function,
and the second part is the
derivative of the inner function.
Let's go ahead and clean this up.
f prime of x is equal to
two times negative one.
That's negative two times one factor
of inverse hyperbolic cosecant of x.
And the denominator would
be the absolute value of x
times the square root
of one plus x squared.
So this would be our derivative function.
Okay, I hope you found
this explanation helpful.
