We're asked to find the derivative function
for the given function.
We have f of x equals natural log of x times the square root
of the quantity four x minus five.
Notice how we do have a composite function here,
so we could let u equal the inner function of x
times the square root of the quantity four x minus five,
but then notice to determine u prime
to apply the chain rule, we would have to apply
the product and the chain rule.
Instead of taking that approach, we are going to use
the log properties shown here on the right
to expand natural log of x times the square root
of the quantity four x minus five.
We'll begin by applying the product property of logarithms
which states the natural log of x times y
equals natural log x times natural log y.
And because we have natural log of x
times the square root of the quantity four x minus five,
we can write f of x as f of x equals natural log x
plus natural log of the square root
of the quantity four x minus five,
but let's write the square root using a rational exponent,
which will give us natural log of the quantity
four x minus five raised to the power of positive 1/2.
And then from here, we can use the power property
of logarithms, which is this last property,
where the natural log of x to the power of y
equals y times natural log x,
which means you can write f of x
as f of x equals natural log x plus 1/2
times natural log of the quantity four x minus five.
And now we can find the derivative more easily
using f of x in this form.
F prime of x is equal to the derivative of natural log x,
which is one divided by x, plus the derivative
of 1/2 natural log of the quantity four x minus five.
Notice this is going to require the chain rule,
where the inner function u is four x minus five.
So the derivative is equal to 1/2
times the derivative of natural log of the quantity
four x minus five, which is equal to
one divided by the quantity four x minus five
times the derivative of four x minus five, which is four.
And let's determine this product.
Before we multiply, notice we can simplify.
Four and two share a common factor of two.
There's one two in two, and two twos in four.
So this gives us f prime of x
is equal to one divided by x,
and then plus two divided by the quantity four x minus five.
So while this is the derivative function,
let's go ahead and add these two rational expressions
by obtaining the least common denominator,
which in this case is going to be
the product of x and four x minus five.
So we'll multiply the numerator and denominator
of the first fraction by four x minus five.
We'll multiply the numerator and denominator
of the second fraction by x.
Notice now we have a common denominator
of x times the quantity four x minus five,
and the numerator is going to be
one times the quantity four x minus five,
which is four x minus five, and then plus two times x,
which gives us plus two x.
Let's write the final derivative function
as f prime of x equals the quantity
six x minus five, combining like terms,
divided by the product of x and four x minus five.
I hope you found this helpful.