Calculus question involving derivatives and fractions
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Calculus question involving derivatives and fractions

[From: ] [author: ] [Date: 11-05-03] [Hit: ]
With the extra factor of 1/3, you get (1/3)(1/4) = 1/12.In general, x^(a/b) is the bth root of x to the ath power.......
So in an example my calculus professor did today, we were doing linear approximation and were looking for the equation of the tangent line via L(x) = f(a) + f'(a)(x-a)

where f(x) = cube-root(x)

and a = 8

So f(a) is obviously 2, and f'(a) would be the derviative of cube-root(8) which should be...

(1/3)(8)^(-2/3)

which, as a fraction, is 1/12. Now, the rest of the problem and the answer doesn't even have anything to do with my question. I'm wondering how you go about converting (1/3)(8)^(-2/3) into fraction form. It's easy enough to just get the decimal answer with a calculator, but my professor just whipped it out on the board and I'm not sure how he got it. I worked the whole thing out using decimals on my calculator and still got the same answer we got in class, so technically it shouldn't matter which method we use. However, I'm currently working on a homework problem that is the exact problem only with different numbers. I'm working it out exactly how we did in the example, but I go to enter the answer (the homework is all online) and I keep getting it wrong. I'd like to understand how to convert a derivative such as the one above into a fraction, because I feel like it would be easier for me to work the problem out using fractions instead of decimals.

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8^(-2/3) = 1/[8^(2/3)] because of the negative exponent. The 2/3 power means take the cube root and then square it. The cubed root of 8 is 2 and 2² = 4. So

8^(-2/3) = 1/(8^(2/3)) = 1/2² = 1/4.

With the extra factor of 1/3, you get (1/3)(1/4) = 1/12.

In general, x^(a/b) is the bth root of x to the ath power.
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