For us to find the first or second
derivative functions given f of x
equals two e to the x times cosine x.
Because f of x is a product of two functions of x,
we'll need to apply the product rule of differentiation
shown here on the right.
Where will the first function be equal to u
and the second function be equal to v.
So in our case, let's set u equal two e to the x,
and v equal cosine x.
Now we can find the first derivative,
and then once we find the first derivative,
we'll differentiate the first derivative
to determine the second derivative.
So f prime of x, is equal to the first function, or u,
which is two e to the x
times the derivative of the second function,
which is the derivative of v,
which gives us a derivative of cosine x with respect to x.
And then plus the second function, or v,
which is cosine x
times the derivative of the first function
which is the derivative of u,
which gives us a derivative of two e to the x
with respect to x.
F prime of x is equal to two e to the x
times the derivative of cosine x,
which is negative sine x.
And then plus cosine x
times the derivative of two e to the x with respect to x,
which is two e to the x.
And now let's go ahead and factor this.
Notice how the greatest common factor
between these two products is two e to the x.
Let's write f prime of x as f prime of x
equals two e of the x times the quantity.
Let's write the quantity as cosine x minus sine x.
Notice how they multiply,
we do still have the same expression.
So here is the first derivative.
And now to determine the second derivative,
we need to differentiate the first derivative
with respect to x.
So to find the second derivative,
we will let two e to the x
be equal to the first function, or u,
and we'll let cosine x minus sine x be the second function
which will be the function v.
So now the second derivative
is equal to the first function or u,
which is two e to the x times the derivative of
the second function or the derivative of v,
which is the derivative of cosine x minus sine x
with respect to x.
And then plus the second function, or v,
which is the quantity cosine x minus sine x,
and then times the derivative of the first function,
or u prime, which is the derivative of two e the x
with respect to x.
And now we need to find the two derivatives.
We have f double prime of x is equal to two e to the x
times the derivative of cosine x minus sine x
with respect to x,
which is negative sine x minus cosine x,
and then plus the quantity cosine x minus sine x
times the derivative of two e the x with respect to x,
which is two e the x.
And now let's go ahead and distribute
and then simplify if we can.
F double prime of x is equal to
negative two e to the x sine x
minus two e to the x cosine x.
And then we have plus two e to the x cosine x
minus two e to the x sine x.
Notice how two of these terms are opposites.
Negative two e to the x cosine x
plus two e to the x cosine x simplifies to zero.
The remaining two terms are like terms.
Negative two e to the x sine x minus two e to the x sine x
is negative four e to the x sine x.
F double prime of x is equal to
negative four e to the x sine x.
I hope you found this helpful.