Here, we're given F of X,
and asked to find F-prime of
X, or the derivative function.
Notice how our function, F of X,
is a quotient of two
functions, so we need to apply
the quotient rule of
differentiation, given here below.
Well, the derivative of function F,
divided by the function
G with respect to X
is equal to G times F-prime,
minus F times G-prime,
divided by G-squared, or we can say
the denominator times the
derivative of the numerator,
minus the numerator times the
derivative of the denominator,
divided by the denominator-squared.
So notice in our case, function F would be
negative three cosine X plus five,
and function G would be five
X to the sixth minus two.
So, for the first step,
we'll go ahead and setup
this derivative formula,
and then we'll find F-prime
and G-prime in the second step.
So F-prime of X
will be equal to
the denominator, which is
five X to the sixth minus two,
times the derivative of the numerator,
or the derivative of negative
three cosine X plus five,
minus the numerator, negative
three cosine X plus five,
times the derivative of the denominator,
or the derivative of five
X to the sixth minus two.
This is all divided by
the denominator-squared,
or five X to the sixth minus two, squared.
Now, we'll find the derivative here,
and the derivative
here, and then simplify.
So we still have five X
to the sixth minus two.
Now, we'll find the derivative of
negative three cosine X plus five.
Well, the derivative of
cosine X is negative sine X,
so this would be negative
three times negative sine X,
or positive three sine X,
and the derivative of a constant
is zero, so that'd be zero,
and then, minus negative
three cosine X plus five
times the derivative of five
X to the sixth minus two.
The derivative of five X to the sixth
would be 30 X to the fifth,
and again, the derivative
of a constant would be zero.
Denominator stays the same.
If we're not required to simplify further,
we could go ahead and stop here,
but I'm going to go ahead and continue.
I'm going to go ahead and
distribute the three sine X here,
and distribute the 30 X to the fifth here.
So, we'll have F-prime of X equals...
Okay, our denominator's
going to stay the same.
Then here, if we distribute
three sine X, we would have
15 X to the sixth sine X,
minus six sine X.
Here, we'd need to be
careful about the signs,
because we're subtracting this product.
Notice, negative three cosine
X times 30 X to the fifth
would be negative 90 X
to the fifth cosine X,
but if we subtract a
negative, that would become
plus a positive so plus 90
X to the fifth cosine X,
and then, 30 X to the
fifth times five, that'd be
150 X to the fifth, but
we're subtracting that,
so minus 150 X to the fifth.
We have no like terms in the numerator,
so we'll go ahead and stop here,
and leave our derivative in this form.
I hope you found this helpful.
