21–30. Derivatives
a. Use limits to find the derivative ...

Question

21–30. Derivatives

a. Use limits to find the derivative function f' for the following functions f.

f(x) = 1/x+1; a = -1/2;5

Accepted Answer

Below there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Find the derivative of the function H of X is equal to 3 divided by x + 2 using the limit definition of the derivative. Awesome. So it appears for this particular problem, we're trying to figure out what the derivative of the specific function of H of X is, and we're asked to figure out the derivative using the limit definition. So with that said, let's read off our multiple choice sensors to see what our final answer for the derivative might be, noting that it states that H of X is equal to some expression. So A is 3 divided by X + 2 to the power of 2, B is negative 3 divided by X + 2 to the power of 2, C is -3 divided by 2 multiplied by X + 2 to the power of 2. And finally, D is -3 divided by X + 2. OK. So first off, in order to solve this problem, we must first define the limit definition of the derivative, and as we should recall a note, the limit definition of the derivative is given by the following expression that H of X is going to be equal to the limit evaluated at H approaches 0 of H X plus. H Minus H of X. All divided by H. So to quickly recap here, so once again, our equation for the limit definition of the derivative for this particular problem since we're dealing with H of X, we could say that H of X is equal to the limit evaluated at H approaches 0 of H X plus H minus H of X all divided by H. So now, our next step is we need to substitute the function. H of X is equal to 3 divided by X + 2 into our limit definition that we just defined together. And we also need to find H of X plus H. So with that said, we need to note that H of X plus H is equal to. 3 divided by X plus H + 2. Which we need to say that this is also equal to, so we're saying 3 divided by X plus H + 2 is all equal to 3 divided by X plus H + 2. Fantastic. So now we can substitute the limit definition. We could substitute in our values, so we could say that H of X is equal to the limit evaluated at H approaches 0 of 3 divided by X plus H + 2. And we could say that this is subtracted by 3. Divided by X plus 2, all divided by H. Fantastic. So now our next step is we simply need to simplify the expression inside of the limit, so we need to combine our fractions. So when we combine our fractions, we will get 3 divided by X plus H + 2, so X plus H + 2. Fantastic. And then we need to take this, and we need to subtract 3 divided by X + 2, and we need to say that this is all equal to 3, multiplied by X + 2 minus 3 multiplied by X plus H + 2. All divided by X plus H + 2, all multiplied by X + 2, and this is all equal to 3 X + 6 minus 3X minus 3H minus 6. All divided by X plus H + 2, all multiplied by X + 2. And this is all gonna be equal to -3H divided by X plus H + 2, all multiplied by X + 2. Fantastic. And we now need to substitute this back into our limit, since now that we found a simplified expression. So when we do that, we will find that H of X is equal to the limit evaluated at H approaches 0 of. -3H all divided by H of X plus H + 2 multiplied by X + 2, which I said H of, but it's H multiplied by X plus H + 2 all multiplied by X + 2. I should be very specific here. Awesome. So now, We need to simplify the limit expression. So when we do that, or in order to do that, as we should recall, is we need to cancel the H in the numerator and the denominator. So when we do that, we will find that H prime. Of X is going to be equal to the limit evaluated at H approaches 0 of -3 divided by X plus H + 2, all multiplied by X + 2. Fantastic. So, Our next step now is we need to evaluate the limit as H approaches 0. So when we do that, we will find that H of X is equal to -3 divided by X + 2 to the power of 2. And that's it. That is our final answer. Hooray, we did it. So essentially when we plug in a value of 0 in for H, we will find when we simplify this once again, this is when we are evaluating our limit as H approaches 0. So we plug in a value of 0 in for our variable h, and when we do that, we will find that our final answer is H X is equal to -3. Divided by X + 2 to the power 2, and you can also pull out the negative sign to be out front of this fraction, or you can have it up on top with the 3, but it's, it's a matter of preference, both are correct. So, looking at our multiple choice answers, our final answer has to be multiple choice answer letter B. Which once again is H of X is equal to -3 divided by X + 2 to the power of 2. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(x) = 1/x+1; a = -1/2;5

Key Concepts

Derivatives

Limits

Difference Quotient

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