Differentiate and simplify as much as possible.
Cube root(5z+6/-9z+3).
The answer should be y'=
(23(-9z+23)^2/3)/9(3z-1)^2(5z+6)^2/3)
So far, I'm stuck at y'=[23/3(5z+6)(3z+1)][((5z+6)^1/3)/((-9z+3)^1/3)]
if you let w be the fraction, then you have
w^(1/3)
which has derivative
1/3 w^(-2/3) dw/dz
Now just use the quotient rule on dw/dz, and you get
69/(-9z+3)^2
That gives you
1/3 * ((5z+6)/(-9z+3))^(-2/3) * 69/(-9z+3)^2
1 * (-9z+3)^(2/3) * 69
---------------------------------------
3 * (5z+6)^(2/3) * (-9z+3)^2
which you can see is the desired answer. I see a (3z+1) in your answer, which is already a typo. Try breaking it up into factors as I did, and you should see things cancel.
using the quotient rule, by first line derivative is
y' = (1/3)[(5z+6)/(3-9z)]^(-2/3) [ (5(3-9z) - (-9)(5z+6) ]/(3-9z)^2
= (1/3)[(5z+6)/(3-9z)]^(-2/3) [ 69/(3-9z)^2 ]
= 23 (3-9z)^(2/3) / (5z+6)^2/3 [ 1/(3-9z)^2 ]
= not getting your answer
let's try the product rule
y = (5z+6)^(1/3) ( 3-9z)^(-1/3)
y' = (5z+6)^(1/3) (-1/3)(3-9z)^(-4/3) (-9) + ( 3-9z)^(-1/3) (1/3)(5z+6)^(-2/3) (5)
= (9/3)(5z+6)^(1/3) (3-9z)^(-4/3) + (5/3)(3-9z)^(-1/3) (5z+6)^(-2/3)
= (1/3)(5z+6)^(-2/3) (3-9z)^(-4/3) [ 9(5z+6) + 5(3-9z)]
= 23(5z+6)^(-2/3) (3-9z)^(-4/3)
check through my steps
looks like Steve and I have the same answer
To simplify the expression cube root(5z + 6) / (-9z + 3), we can start by rationalizing the denominator.
First, let's simplify the expression inside the cube root:
Cube root(5z + 6) / (-9z + 3)
Now, let's rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator, which is (-9z + 3).
[(Cube root(5z + 6) / (-9z + 3)) * ((-9z + 3) / (-9z + 3))]
This gives us:
Cube root((5z + 6)(-9z + 3)) / (-9z + 3)(-9z + 3)
Next, let's simplify the numerator:
Cube root((5z + 6)(-9z + 3))
To simplify this product under the cube root, we need to factor out the common factors from both terms.
Cube root(-45z^2 -15z + 18z + 6)
Cube root(-45z^2 + 3z + 6)
Now we can rewrite the expression as:
(Cube root(-45z^2 + 3z + 6)) / (-9z + 3)(-9z + 3)
At this point, there's not much we can simplify further without using numerical values for z. However, if you're looking for an expression similar to the provided answer, you can try expanding the numerator and denominator:
(-45z^2 + 3z + 6) = (-9z + 3)(5z + 2)
So the expression can be further simplified as:
(Cube root((-9z + 3)(5z + 2))) / (-9z + 3)(-9z + 3)
But please keep in mind that this is just another way of representing the original expression; it may not exactly match the answer you provided.