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Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$ f(x) = x^2 - 2x^3 $

$2 x-6 x^{2}$

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 8

The Derivative as a Function

Limits

Derivatives

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There's probably twenty five of this to a calculus eighth addition. Section two point eight find the dirt of the function using the definition of curative state, the domain of the function and the domain of its door. The function is of X equals X squared minus to execute and the definition of the derivative is shown in green. We're going to use this definition for the function given and find the paramedics in the limits and the function is X squared minus two x cubed s o. The first term is dysfunction evaluated at X plus age. So it's explicit age quantity squared minus two times the quantity x each Cuban Ah, and then subjecting the function x squared minus to execute Thank you in this old age andan except will be to expand the numerator. This is a binomial squared which will become X squared plus two x h plus h squared. When off oil, the next term is explicit, that quantity cubed and that will be four terms executed plus two xx goes three x squared age plus three exchange squared plus age cubed on that almost by native to gives us things to x cubed find a six x squared age minus six. X h squared mine too. H cube. And then we subtract function right, Miss X. Cubed Plus to execute on this all developer H Okay. Ah, we have a next square term here that cancels out with, um, this negative X squared term. We have a to execute two term that cancels that with this, a positive to execute a term. And then the remaining terms each have one h, at least in the numerator, which can cancel that with the one age ended denominator. So here we shown one age being counseled from each remaining term. And these are many terms that we now ah, rear ain't simplified in the next line, all that's left over is too x wass each minus six x squared minus six eight x h minus two h squared. And although we have a lot of terms here, when we evaluate the limit, age approaches zero each h term. Each term with an H vanishes. That's H negative six X h and native to X squared, leaving only two x my specs squared minus x X squared as the remaining function. And this isn't need the derivative of the function after So the function of X X squared minus to execute its derivative is de function to X minus six x squared which we found isn't the definition and the domain of both of these. Since there are no meals, it is the domain off all real numbers from negative infinity two hundred eighty.

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