Question

differentiate cube root of x^2 (2x - x^2)

Answer 1

Answer:

Explanation:

What we have here is a function within a function; x2+2x−1 is under the radical (√) sign. That means we have to use the chain rule to differentiate, which says that you take the derivative of the "inside" function (in this case x2+2x−1) and multiply it by the derivative of the whole function.

Begin by finding the derivative of x2+2x−1. Using the power rule, the derivative is 2x+2. Now onto the whole function. Note that we can write √x2+2x−1 as (x2+2x−1)12. That means we can again apply the power rule:

ddx(x2+2x−1)12=12(x2+2x−1)12

Now we can multiply this by the derivative of the inside function, which we found as 2x+2. Performing this operation yields:

12(x2+2x−1)12⋅2x+2=2x+22(x2+2x−1)12

Finally, look for any ways to simplify the problem. We see that there is a 2 in the denominator - is there any way we can get rid of it? In fact, there is by factoring out a 2 from the numerator; take a look:

2(x+1)