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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) = mx + b $

$f^{\prime}(x)=m, \mathbb{R}, \mathbb{R}$

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Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

to some number, M times X plus some number of big. The domain of this function uh is any X value uh that uh where this function would be defined? Well you can see that dysfunction would be defined no matter what number you plugged in, correct? So the domain is the set of all real numbers or the entire X axis. Now we want to calculate the derivative of this function using the limit definition of derivative. So F prime of X is going to be to limit has h approaches zero of this function evaluated at X plus H. So the limit of F X plus H minus F. A bex. All divided by each. This is the limit definition of the derivative. The derivative F prime of X is the limit of F of X plus H minus F of X. All divided by H has H approaches zero. Now if we're going to take the limit of this function As a church approaches zero first we have to determine what is F of X plus H. F of X is M times X plus B. So F of X plus H plug in X plus H. Where you see X. So M times X plus H. Uh And then plus B. That is our F. Of X plus H. And then we have to subtract F of X. So we're gonna subtract this F of X function. So we're going to be subtracting Mx. And then we're also going to subtract B. And all of this gets put over H, mm. So doing just a little bit of algebra, we had to distribute this multiplication by the M. So M times X plus M times H. Ah plus B. To attract Mx should track B and all that is over the H. Some of these terms are going to cancel out, make our work a little bit easier. So M. X. Subtract mx cancels plus B. Subtract B cancels. We have M times H over H. Now the H is canceled. And so the limit of this expressionist H approaches zero is just M. So F. Prime of X is equal to M. That's the derivative and the domain of this derivative. Well, since the derivative is a constant, uh it doesn't matter what value X. Is. So. Extra be any real number. So the domain is any real number along the X axis.