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Find the derivative of the function.$ f(x) = \frac {1}{\sqrt [3]{x^2 - 1}} $
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00:27
Frank Lin
01:56
Heather Zimmers
Calculus 1 / AB
Chapter 3
Differentiation Rules
Section 4
The Chain Rule
Derivatives
Differentiation
Missouri State University
Oregon State University
University of Nottingham
Lectures
04:40
In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.
44:57
In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.
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All right here we have the function on left. F of x equals 1 over cube root of x, squared minus 1 point i'm going to go ahead and rewrite it. So the x, squared minus 1 is in the numerator on top, which will bring it to the minus 1 third power now notice that this is a composite function. The inner function is x, squared minus 1 and the outer function is power up to the minus 1. Third, so when we have a composite functions, we can use chain rule, and i showed it to you on the right side chain. Rule f of x is if, for example, f of x is a composite g of h of x, then f, prime of x, is going to be the derivative of g with h of x. Still the argument times h, prime of x. In so, let's go ahead and do this so in this case we notice that we have an inner function in the outer function. The inner function is our h of x and are basically taking us to the minus lether power is what g of x is doing? Okay, so let's go ahead and solve this for the derivative f prime of x, then is so what we're going to do. Is we take the outer function, which is to the minus 1 third power, an we're goin to do power rule and we're going to leave the argument the same so this argument inside that stays the same. We bring the minus 1 third down as a multiplier for power rule and then we subtract 1 from the exponents. So we get minus 4 thirds okay, so the part we did so far is just this part. Now we're going to multiply by the derivative of the inside, which is a derivative of h in our little example. So the derivative of the inside derivative of x, squared minus 1, is just 2 x, minus 0 point okay, so we're really done other than any kind of clean up. Let'S see if we can do any kind of clean up, that's helpful! I'M going to go ahead and, let's see it looks like we have, can put a minus in front. We have a 2 x that stays on top. The denominator is 3. I'M going to go ahead and put this x squared minus 1 back on the bottom. Just so i can make it a positive exponent and i think that's why about all we can do so. There is our solution for our derivative for that help to have a wonderful day.
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