We're asked to find the derivative function
of each given function.
First we have g of x equals negative eight square root of x
times sine x.
Because g of x is a product of two functions of x,
we need to apply the product rule of differentiation
to find the derivative.
The product rule stated to the right
is the derivative of u times v with respect to x
is equal to u times v prime plus v times u prime
or the first function
times the derivative of the second function
plus the second function
times the derivative of the first function.
We also need to write this square root of x
using a rational exponent.
So let's first write g of x
as g of x equals negative 8x to the power of 1/2
times sine x.
Remember the square root of x is equal to x to 1/2
because the exponent on x is one under the square root
and the index for the square root is two.
So now from here, g prime of x, applying the product rule,
is equal to the first function
of negative 8x the power of 1/2
times the derivative of the second function,
which is a derivative of sine x
plus the second function of sine x
times the derivative of the first function,
which is a derivative of negative 8x the power of 1/2.
And now let's find the two derivatives.
G prime of x is equal to negative 8x the power of 1/2
times the derivative of sine x
with respect to x is cosine x.
Then we have plus sine x
times the derivative of negative 8x the power of 1/2
with respect to x,
which is equal to negative eight
times the exponent of 1/2
times x the power of 1/2 minus one
which gives us an exponent of negative 1/2.
And now let's simplify.
We have g prime of x equals negative 8x
the power of positive 1/2 cosine x.
Then we have plus negative eight times 1/2 is negative four.
So let's write this as minus 4x
the power of negative 1/2 sine x.
So this is the derivative function.
Let's go ahead and write it using positive exponents.
We can think of the second product
as a fraction with the denominator of one
and then we can move x the power of negative 1/2
to the denominator,
which will change the sign of the exponent.
So we have g prime of x
is equal to negative 8x the power of positive 1/2 cosine x
minus four sine x divided by x the power of 1/2.
This would be one way
we might see the derivative function.
Another way would be to rewrite x the power of 1/2
as the square root of x.
So we can also express the derivative as g prime of x
equals negative eight square root x cosine x
minus four sine x divided by the square root of x.
Either of these last two forms
would be the most common way
to express the derivative function.
And now let's find the derivative of f of x.
Again, let's write the cube root of x squared
using a rational exponent.
So we have f of x equals negative 6x
to the power of 2/3 cosine x.
And again we'll have to apply the product rule
to determine f prime of x.
F prime of x is equal to the first function
of negative 6x the power of 2/3
times the derivative of cosine x with respect to x
plus the second function of cosine x
times the derivative of the first function
which is a derivative of negative 6x the power of 2/3
with respect to x.
And again, let's find the two derivatives.
F prime of x is equal to negative 6x the power of 2/3
times the derivative of cosine x, which is negative sine x.
And then we have plus cosine x
times the derivative of negative 6x to the 2/3,
which is equal to negative six times 2/3
times x the power of 2/3 minus one, which is negative 1/3.
And now we simplify.
We have f prime of x is equal to,
well here we have a negative times a negative,
so we have positive 6x to the 2/3 sine x.
And then plus negative six times 2/3
is equal to negative four.
So let's write this as minus four,
instead of plus negative four,
x the power of negative 1/3 cosine x.
Let's also write this using a positive exponent.
So we'll move x the power of negative 1/3
down to the denominator.
We have 6x the power of 2/3 sine x minus four cosine x
divided by x the power of positive 1/3.
Let's also express the derivative using radicals.
So in both cases,
because the denominator of the rational exponent is three,
we have two cube roots.
So we have f prime of x equals six
times the cube root of x squared sine x
minus four cosine x divided by the cube root of x the first
or just the cube root of x.
These last two forms would be the most common way
to express f prime of x.
I hope you found this helpful.