whats the derivative of ((2/3x)^2)-(x/6)?

Answer 1

I agree with both of your methods to find the derivative, however I have to point out that even before you pull out fractions before differentiation, you have to use spread the exponent among the variables and constants, in this case (2/3x). Since the (2/3) is included in the parenthesis for the exponent, it, too, is squared.

Basically, the answer is actually,

8/9x-1/6

Answer 2

I don't know if your teacher taught you this method yet, but derivatives like this one are simple to solve. If you don't know this method, it is very easy to do, and a good way to check your work:

You multiply the coefficient of each term by its respective exponent, and lower the exponent by one power.

So you have:

(2/3X)^2 as the first term.

You multiply 2/3 (the coefficient) by 2 (the exponent) and decrease the exponent (2) by one, so it becomes 1, so you now have as the first term: 4/3X

You also have:

1/6X as the second term.

You multiply 1/6 by the exponent (1) to get the new coefficient, 1/6 (it is the same as the last because the exponent equals 1.

and you decrease the exponent (1) by one to get 0, so for the second term, you get (1/6)

Putting it together, you get

(4/3)x - (1/6)

Answer 3

First step is to do the exponent. Then use the chain rule to find the derivative of what's under the exponent:
You should have
2*(2/(3x))*[derivative of 2/(3x)] - 1/6

Second step
find the derivative of 2/(3x) and substitute it into the above.

Answer 4

Remember: d/dx x^n = n x^(n - 1)

Also, multiplicative constants like 2/3 or 1/6 are pulled out before differentiation. So:

d/dx ((2/3)x^2) - (x/6) = (4/3)x - (1/6)

Answer 5

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