We want to find the derivative of f of x
equals x times hyperbolic sin of x,
plus two times hyperbolic co-sin x.
We'll notice how this first term's applied
to functions, so we'll have
to apply their product rule
in order to find the
derivative of this term here.
So we take a look at
our derivative formulas.
We'll need their product rule.
Stated here...
As well as the derivative
formulas involving
hyperbolic sin x and hyperbolic co-sin x,
which would be these two formulas here.
So let's start by setting
up our derivative.
F prime of x,
can be equal to the
derivative of the first term,
which again requires the product rule.
So the first function is x and the second
function is the hyperbolic sin function.
So the derivative of
this term would be the
first function, x, times
the derivative of the
second function, which
is hyperbolic sin x.
Plus the second function
or hyperbolic sin x,
times the derivative
of the first function,
which will be the derivative of x.
And then plus the derivative of two times
hyperbolic co-sin x.
So notice how because this
derivative require their
product rule, we didn't
find any derivatives
on this first step, we just
have their product rule,
and now we'll go back
and find the derivatives.
Here, here, and here.
So f prime of x can be equal to x times
the derivative of hyperbolic sin x.
Notice how this derivative
formula here does
include the chain rule, but
since u would be equal to x,
and u prime would be equal to one,
this derivative does not
require the chain rule.
So our derivative would just be hyperbolic
co-sin x plus hyperbolic sin x,
times the derivative of
x, which is just one.
Plus the derivative of two
times hyperbolic co-sin x.
Which again, does not
require the chain rule,
so our derivative would just
be two times hyperbolic sin x.
So this our derivative, but
we can go ahead and combine
these two terms here,
because they're like terms.
So f prime of x is equal to
three hyperbolic sin x,
plus x hyperbolic co-sin x.
And that'll do it for this example.
I hope you found this helpful.
