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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^4 $

$4 x^{3}$

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 8

The Derivative as a Function

Limits

Derivatives

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this is problem number thirty one of this to re calculus eighth edition, section two point eight Find the dirt or the function using the definition of terror of you state the domain of function And did I mean if it's true, f of X equals X fourth and the definition dirt and we're going to use is probably equals. Limited's expert today of the function minus F b developed by the quantity. Explain this, eh? So we're going to planet here distributive and see what the function should look like. So we have a function next for ninety eight of the fourth or a represents just any number in the domain of the function. Um, if we factor the numerator we can factor is a difference of squares X squared, plus a squared, X squared minus squared, divided by experience saying, And since there's another difference of squares in the numerator, we can do it once more. This eight expert with a score term remained, stays the same, and then the other term X squared minus a squared can be split into. Experts say time's explain. It's a on the right experience. He now we see that the experts a terror councils on with the denominator, we're left with just X squared, plus days where the quantity multiplied by extra thie and exit purchased. A function approaches a squared plus eight squared times a plus aim, which is to a squared Time's too, eh? Which is for a cube. Now, if we generalize the solution for any A for any X value of a he, we say the derivative function is equal to four x cubed, and that is definitely our derivative. If we take a look at the function original function except forthe and the dirt a function of forks cubed these air both polynomial functions and therefore their domain is all real numbers. They could have been any to infinity.

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