To determine the derivative
of f of x equals two cosine x sine x,
we need to recognize this as a product
of two differentiable functions.
So the first function
will be two cosine x,
which we'll call f, and the
second function will be sine x,
which we'll call g.
The product rule states
the derivative of f times g
is equal to f times g
prime plus g times f prime,
where the derivative is
equal to the first function
times the derivative of the
second plus the second function
times the derivative
of the first function.
So we'll have f prime of x is equal to
the first function, which is two cosine x
times the derivative
of the second function,
the derivative of sine x is cosine x,
plus the second function, which is sine x
times the derivative of the first function
and the derivative of two cosine x
would be negative two sine x.
So we have f prime of x is equal to
two cosine-squared x.
This is going to be
minus two sine-squared x.
So we applied that product
rule correctly here,
and there aren't any like terms,
but in many textbooks, this would not be
what you'd find in the back of the book
as the correct answer.
While this derivative function is correct,
it can be simplified by
doing a trig substitution.
Notice how these two terms
have a common factor of two.
So if we factor out the
two, we would have two
times the quantity cosine-squared
x minus sine-squared x.
Now in this form, you may
recognize the identity
that we can use.
Be careful, this is not a
sum, so it's not equal to one,
but it is equal to cosine two x,
using the double angle
identity here for cosine.
So we can rewrite this derivative as
two times cosine two x.
And more that likely, this
is the answer you'd find
in the back of a textbook.
So in determining derivatives
involving trig functions,
don't panic if your answer doesn't match
what a textbook says is correct,
because you'll find they'll
perform a trig substitution
to simplify the function further.
I hope this was helpful.
