We want to find the derivative
for the given function.
Notice how we do have a composite function,
where the outer function is a natural log function,
and the inner function is the product
of the three inner functions.
In this case, to find f prime of x, it'll be much easier
to first expand the logarithm on the right,
using the log properties.
We'll begin with the product property of logarithms,
which is natural log of the quantity x times y
equals natural log x plus natural log y.
Notice in our case,
we have the natural log involving three products,
where we have natural log of x to the eighth
times the cube of x plus 7
times the fifth power of 2x cubed plus 1.
Which means you can write f of x
as f of x equals a natural log of x to the eighth
plus natural log of the third power of x plus 7
plus natural log of the fifth power of 2x cubed plus 1.
And now we can continue expanding
by using the power property of logarithms,
which is a third property,
natural log of x to the power of y
equals y times natural log x.
So we can move the x pointer
to the position of the coefficient,
which means we can write f of x
as f of x equals the natural log of x to the eighth
is equal to 8 times natural log x
plus natural log of the cube of x plus 7
is equal to 3 times the natural log
of the quantity x plus 7
plus natural log of the fifth power
of the quantity 2x cubed plus 1
is equal to 5 times natural log
of the quantity 2x cubed plus 1.
Now that we've expanded the natural log on the right,
we can now more easily determine the derivative function.
f prime of x is equal to the derivative
of 8 natural log x,
which is equal to 8 times the derivative
of natural log x, which is 1 divided by X.
Notice how, because the inner function is just x,
the chain rule is not needed,
however it's never wrong to apply the chain rule.
Notice, in this case, if we let u equal x,
u prime is just 1.
And then we have plus the derivative
of three natural log of the quantity x plus 7,
which is equal to three times the derivative
of natural log of the quantity x plus 7.
Again, if we're not sure
as to whether we need the chain rule,
let's go ahead and apply it anyway.
Notice here, the inner function u is x plus 7.
And again, notice here u prime is just equal to 1,
and therefore the chain rule is not needed.
The derivative of natural log of the quantity x plus 7
is just 1 divided by the quantity x plus 7.
And then we have plus the derivative
of 5 natural log of the quantity 2x cubed plus 1.
Here we do need to apply the chain rule
where the inner function u is 2x cubed plus 1.
So we have plus the derivative
of 5 natural log of the quantity 2x cubed plus 1,
which is equal to 5 times the derivative
of natural log of the quantity 2x cubed plus 1,
which is equal to 1 divided by u times u prime,
or 1 divided by the quantity 2x cubed plus 1 times u prime,
where u prime is the derivative of 2x cubed plus 1,
which is 6x squared.
Let's go ahead and simplify.
We have f prime of x equals 8 divided by x
plus 3 divided by the quantity x plus 7
plus, for the third product,
the numerator is going to be 5 times 6x square,
which is 30x squared,
divided by the quantity 2x cubed plus 1.
And we'll go and leave the derivative function in this form.
I hope you found this helpful.