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Flx- h)_ flx) (6 points) Use the limit definition of derivative ( f" (x) lim Ito find the derivativeof flx) = X-5...

Question

Flx- h)_ flx) (6 points) Use the limit definition of derivative ( f" (x) lim Ito find the derivativeof flx) = X-5

flx- h)_ flx) (6 points) Use the limit definition of derivative ( f" (x) lim Ito find the derivative of flx) = X-5



Answers

Use the limit definition of the derivative (algebraic method) to confirm the statements. The derivative of $f(x)=-3 x^{2}-5 x$ is $f^{\prime}(x)=-6 x-5$.

We have problem number 16 effects, that is function is given us eight minus one were five x. We need to find the deputy by using the limit process so according to the formula, if Dash X will be equal to limit Delta X tends to zero f x plus tilt X minus FX by deal Tex Okay, now let us start. Our question will be plugging in. Experts will text in place effects in this equation. It will be X minus one by five x plus Delta X minus FX simply eight minus one by five x They would battle decks, so this is limit. Delta extends to zero eight minus one by five x minus one by five Little tax minus eight plus one by five x Divide by Dale Tax So eight and minus it Minds on by five x one by five weeks will get canceled out so we will be having limit. Delta extends to zero minus one by five Bill tax by little tax. So this is minus one by five. So we have different Children. Derivative aftershocks equal toe minus one by five. Thank you so much.

In this exercise, we want to find the derivative of F f of X equals three X minus 12 soldiers. We use a limit definition of derivative with a limit of H is approaching zero of F of X plus a choice f of X over H You put an explosive H we get three X plus three H minus 12. We subtract out after backs, which is three X minus 12 and divided by h. We see that we can cancel out our three x case along with three. Ex Castle is 12. Castle is 12 and all we're left with is three h over H which cancels to giving us that the derivative of X equals three.

Were given Effa Becks in this problem is defined as three X squared. And the whole premise is that the derivative Is if you do the limit definition, so it's a limit as h approaches zero and this I'm just copying, you can find this off the internet after the x minus h minus F of X. All over H. And the whole thing is um this three X squared can substitute in for that F. A bex. You don't even need parentheses because you can just leave that minus there. You leave the H there, Leave this limit limit as H approaches zero. But then you also want to substitute this X plus H in for this X. And that's going to be the bulk of this work. Is because as you simplify that you have to really foil out or distribute because as a reminder X plus H squared is the same thing as X plus H times X plus H. So it's X squared plus two X H plus H squared. So as you distribute that three into the problem, what you should notice is the three X squared will cancel out with that three X squared. And of everything else that we have, I'm assuming you can do some of this math in your head, you would have a six X. H. That you could factor out the H With this other three H. Um And double check my arithmetic although it is correct but now you can cancel out these ages and you can plug in zero and for this H. So you're left with six X plus three times zero. Well three times zero is still zero. Uh and add that to six action. You get six X, which is the right answer.

So what we want to do here is we want to find the derivative of F of X which is equal to five minus 2/3 X. Using the definition of the derivative which I have provided right here. So sorry about that. So what we're gonna do is we're going to use this definition to help us. So that means that we write down the limit as delta X approaches zero of F of X plus delta X. Well, what that means is that for our function F of X? We write down anywhere. I see an X. I'm gonna plug in a X plus delta X instead. Okay, so that means that it is five minus to over three X plus delta acts okay, minus affects. And what is F of X 5 -2/3 X. And all of that is being divided by delta X. Okay, so we're gonna have to distribute something's cancel some things out that gives us the limit. As Delta X approaches zero of 5 -2/3 X -2/3. Delta actually put in parentheses minus five plus to over three X. Because that negative of course distributes across and all that is still being divided by delta X. Well some things are going to cancel out. So we have a plus five and the negative five. So those are gone and then we have a negative to over three X at a positive two or three X. So those are gone. And so what we're left with is the limit. As Delta Acts approaches zero of -2/3 times delta X divided by delta X. And you don't have to worry about the you know, fractions within fractions. All you have to worry about is that you have a delta X at the top of this uh fraction and another delta X at the bottom. And so that means that those delta X is disappear. Okay? They cancel out. And you're left with the limit As Delta X approaches zero of negative 2/3, and the limit of a constant is just a constant, Which will be negative to over three. And so that is our answer.


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