Expert: Scotto - 5/24/2010

Question
I have a problem that I cannot seem to solve.  Can you please help me to understand it?  The problem is:

y = cuberoot(9x^2+4)  

I need to use the General Power Rule to find the derivative of this function.  I've tried so many times, but the more I try the more confused I get.  Can you help?

Thanks,
Leslie

Answer
Take each of the cubes and solve.
Note that y³ = 9x² + 4.
Now 1³ = 1, and since x² is always positive, this can never be.
Take 2³ = 8, and note that this can never be.

Next is 3³ = 27, so if we take 27 = 9x² + 4, that says 9x² = 23, and x² = 23/9.
That might be a solution, but lets look for one with no squareroot.

Next we try 4³ = 64, so if we take 64 = 9x² + 4, we get 9x² = 60, so x² = 60/9, so x = √60 / 3,
which still involves a squareroot.

At 5³ = 125, 125 = 9x² + 4, the 9x² = 121, so x = √(121/9) = 11/3.
Now here, x is not an integer, but it is real.

Let me use Excel for one moment ... there, now I'm done.
For all values of y between 1 and 100, that is the only one I found that comes out to a quotient.
Note that the only thing it could be in was 1/3's since there is a 9 on the bottom.

Now I went a little farther and found that 225^3 = 11,390,625.
When 4 is subtraced, the answer is 11,390,621.
When this is divided by 9, the answer is almost 1125, which is 35²,
but the answer at this point is really 1124.999802, which is only -0.0002 away from 35².

Now if I look a litttle farther, I get 2,500.
The cube is 15,625,000,000.
When 4 is subtracted, the result is 15,624,999,996.
When that is divided by 9, the result is 1,736,111,111.
When the squareroot is taken, it is found to be 125,000/3.
However, due to the size of the numbers involved, this number goes beyond the accuracy of this computer.