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Find the $ n $ th derivative of each function by calculating the first few derivatives and observing the pattern that occurs.(a) $ f(x) = x" $ (b) $ f(x) = 1/x $
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02:00
Frank Lin
Calculus 1 / AB
Chapter 3
Differentiation Rules
Section 1
Derivatives of Polynomials and Exponential Functions
Derivatives
Differentiation
Missouri State University
Harvey Mudd College
Baylor University
University of Michigan - Ann Arbor
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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he It's clear. So in new Marine here. So we're gonna find the first few derivatives you have ever X is equal to X to the end power when we have our first derivative, which is X and times X to the n minus one power. We have our second derivative, which is n times and minus one X to the N minus two. And we see that, uh, to the then power derivative is equal to and times and minus one times and minus two all the way. Tow n minus and close one times X to the n minus. And and this becomes equal to in factorial For part B, we're gonna do the same thing. Just write a couple of derivatives to see the pattern. X Square becomes next for the second er, bit of negative one negative too. Next to the negative three years power. And then we have, uh, then derivative, which is equal to negative one times Negative too, comes negative three all the way to negative and times X to the negative and plus one power which is equal to negative one to the end. Power factorial over X to the end, plus one
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