We're asked to find the derivative
of each function.
Let's begin with f of x
equals x to the power of negative five times e to the x.
Because we have a product,
we need to apply the product rule of differentiation
to find f prime of x.
So beginning with the given function,
for the purpose of the formula,
we will let the first function be u
and the second function be v.
In this form, the derivative of u times v with respect to x
is equal to u times v prime plus v times u prime
or the first function
times the derivative of the second function
plus the second function
times the derivative of the first function.
Which means f prime of x
is equal to the first function, or u,
which is x to the power of negative five
times the derivative of the second function, or v prime,
which is a derivative with respect to x of e to the x.
And then plus the second function, which is v,
times the derivative of the first function, or u prime,
which is the derivative of x to the power of negative five.
And now let's find the two derivatives,
f prime of x is equal to x to the power of negative five
times the derivative of e to the x is e to the x.
And then we have plus e to the x
times the derivative of x to the power of negative five,
which is negative five
times x to the power of negative five minus one,
which is negative six.
Simplifying, we have x
to the power of negative five e to the x,
and then we have minus 5x
to the power of negative six e to the x.
So this is one way we can express the derivative,
but there are several other ways.
For example, begin factoring out the greatest common factor,
which would be x to the power of negative five e to the x.
So this also shows various forms of the derivative function.
F prime of x in factored form
would be x to the power of negative five e to the x
times the quantity one
minus 5x to the power of negative one.
If we distribute,
we still have x to the negative fifth e to the x
minus 5x to the negative sixth e to the x.
And then one more way we could express this
using positive exponents.
X to the power of negative five e to the x
is equivalent to e to the x divided by x to the fifth.
And then we'd have times the quantity one
minus five times x to the power of negative one
is equivalent to five divided by x.
So depending on your materials,
you may find the derivative in any of these three forms.
And now let's find g prime of x.
Again, notice g of x is a product.
For the purpose of our formula,
we will let the first function be equal to u
and the second function be equal to v.
G prime of x is equal to the first function, or u,
times the derivative of the second function, or v prime,
which is the derivative of e to the x with respect to x.
Plus the second function, or v,
times the derivative of the first function, or u prime,
which is the derivative of 3x squared minus two
with respect to x.
And now let's find the two derivatives.
G prime of x equals the quantity 3x squared minus two
times the derivative of e to the x is e to the x.
And then we have plus e to the x
time the derivative of three x squared minus two
with respect to x,
which is 6x.
So this is the derivative function
and we could probably leave it in this form here.
But let's distribute the e to the x.
So we could say g prime of x
is equal to 3x squared e to the x
minus two e to the x plus 6x e to the x.
We could also express the derivative in this form
or we could factor out the greatest common factor
and write the terms in descending order,
which would be e to the x
times the quantity 3x squared
and then plus 6x and then minus two.
So once again,
we can express the derivative in several forms.
We can express the derivative in this form,
this form or this form.
I hope you found this helpful.