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Numerade Educator

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Problem 10 Medium Difficulty

Find the derivative of the function.
$ f(x) = \frac {1}{\sqrt [3]{x^2 - 1}} $

Answer

$-\frac{2 x}{3\left(x^{2}-1\right) \sqrt[3]{\left(x^{2}-1\right)}}$

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Video Transcript

here's her function and it's a composite function, so it's one function inside another, so we're going to find its derivative using the chain rule. And before I do that, I'd like to rewrite this one. Having a cube root on the bottom is like having an X to the negative 1/3. So we have X squared minus one, all to the negative 1/3 power. Okay, that's going to be easier to differentiate. So now for the derivative, According to the chain rule, we start with the derivative of the outside function, and that would be the negative 1/3 power function. So we bring down the negative 1/3 and we raise the inside to the negative 4/3 because we need to subtract one to get the new power. Now we're going to multiply by the derivative of the inside and the derivative of the inside would be the derivative of X squared minus one. So that would be two X. So we have our derivative, and now we just need to simplify. So what we're going to do is combine the negative 1/3 and the two X and then we're also going to get rid of our negative exponents. So that gives us negative two X over three times the quantity X squared minus one to the positive 4/3. Now, another thing we would like to do is change of actor radical form. So remember that a 1/3 power is a cube root. So we have negative two X over three times the cube root of X squared plus one minus one to the fourth. Okay, we can simplify that radical by pulling an X squared minus one out of it since we have four of them in there. So our final answer is negative two x over three times a quantity X squared, minus one tens. A cube root of X squared minus one.