We want to find the derivative of f of x
equals three times hyperbolic sine of x.
I'm going to show how to find
this derivative two ways:
one way by just using the
basic derivative formula
provided here, but also by
converting the given function
to exponential form and
then finding the derivative
as an exponential function.
Notice how this derivative formula
does include the chain rule
where the inner function
would be equal to u.
And therefore, we have the derivative
of hyperbolic sine of u with respect to x
is equal to hyperbolic
cosine of u times u'.
But for our given function,
since the inner function
would just be u equals x,
u' would be equal to one
which means you don't need
to apply the chain rule.
Therefore, f' of x is equal to three
times the derivative
of hyperbolic sine of x
which is equal to hyperbolic cosine of x.
So this would be our derivative function.
But again, I also want to show
how to find this derivative
by writing f of x as
an exponential function
and finding the derivative
in exponential form.
So f of x equals three
times hyperbolic sine of x
would be the same as three times
e to the power of x
minus e to the negative x
divided by two.
So let's go ahead and rewrite this
as three halves times e to the x minus
e to the negative x.
And let's find the
derivative in this form.
So f' of x would be equal to three halves
times our derivative of e to the x
which is just e to the
x minus the derivative
of e to the negative x
which would require the chain rule
where the inner function
is equal to negative x.
So the derivative would
be e to the negative x
times the derivative of negative x
which is negative one.
So this would be three
halves times e to the x.
Now this becomes plus e to the negative x
which now we can write
as three times e to the x
plus e to the negative x divided by two.
In this form, hopefully
we recognize that this
is equal to hyperbolic cosine of x,
and therefore, of course,
this derivative is the same.
It would be equal to three
hyperbolic cosine of x.
Of course, normally to
find the derivative,
we'll just apply the
collect derivative formula.
But I did want to show at least once
if we write the hyperbolic
function in exponential form
and find the derivative,
of course the derivative
would be the same.
I hope you found this helpful.
