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) = 4 + 8x - 5x^2 $
Okay here we have the function F of x equals four plus eight x minus 58. Uh Four plus eight X minus five X squared. Uh The domain of a function uh are all the values uh that can be inputted into the function. So basically all values of X for which is a function would be defined. Dysfunction is going to be defined for every real value effects. So we're going to say the domain is the entire X axis or all real numbers. Now we have to find the derivative of this function uh Using the definition of the derivative. So F prime of X is going to be to limit As a judge approaches zero of F, evaluated at X plus H minus F. Evaluated at X. All divided by H. Yeah, As a church approaches zero, so define F prime of X. We have to take a limit of this function evaluated at X plus H minus the value of the function at X. Given right here. All divided by H as H approaches zero. So what we're going to do first is we're gonna substitute X plus H everywhere. And you see ex uh then we're going to subtract the function F fx written here, divided all by H. Do a little bit of algebra and take the limit as H approaches zero. So f evaluated X plus H will be four plus eight times X plus H uh minus five times X plus H squared. And then we have to subtract ffx. So we're gonna subtract each of these terms in ffx. So we're gonna subtract before subtract the eight X. And then we're also going to subtract the minus five X. Squared. Which turns into a plus five X squared. All that is divided by eight. And were taken to limit as H approaches zero continuing simplifying this with a little bit of algebra. Uh This eight is going to be multiplying the X plus eight so we'll distribute it eight X plus eight H. X. Plus H. Squared. Uh This has to be uh X plus H. Is multiplying by itself multiplying itself because it's being raised to the second power. So we'll use foil to do X plus H. Times X plus H. Okay so after a little algebra d eight times X plus age was a let's fix this. Eight X plus eight H let's make distant pecs. Uh And then the X plus H squared was X squared plus two hx plus H squared uh ties by the five. You get minus 585 X squared minus 10 times X. H minus 58 squared. And then we subtract minus four minus eight X plus five X squared all of that being divided by h. Now things will begin to simplify a little bit uh for uh minus four cancels plus eight X minus eight X cancels uh minus five X. Square plus five X square cancels then. So now we have a little bit more simpler expression here. Ah So this is equal to the limit As a church approaches zero of eight H Uh -10 X. H. Uh minus five H squared. All divided by each. Now each one of these terms in the numerator uh can individually be divided by H. So were taken to limit as h approaches zero eight H. Divided by H. Is just eight. Subtract uh 10 X. H. Divided by H. Is 10 X. Uh And then minus 58 square divided by H is just minus five H. And as we take the limit of this expression as h approaches zero uh five H is going to approach zero and so we're going to be left with uh eight minus 10 X. And so change of color. Uh The limit of this expression as H approaches zero since this term here will approach zero. We're left with eight minus 10 X. Uh So F prime of X. The derivative of our function F of X. F prime of X equals the limit of this expression is H approaches zero which is eight subtract 10 X. So there is F. Prime of X. And last but not least if we want to talk about what is the domain of F prime of X. What values of X can be plugged in? You can see that any real number can be plugged into this function