Find f′(x), f′′(x), and f′′′(x) for the following functions.

Question

f(x) = (x² - 7x - 8) / (x + 1)

Accepted Answer

Hello there. Today we're going to 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. Determine the 1st, 2nd, and 3rd derivatives of the function F of X is equal to x 2 + 3X minus 10 divided by X + 5. Awesome. So it appears for this particular problem we're asked to solve for 3 separate answers. We're trying to figure out the 1st, 2nd, and 3rd derivatives of this function of F of X. So those are our three separate answers, the 1st, 2nd, and 3rd derivatives of this function F of X. So, with that said, our first step in order to solve for this particular problem, as we should recall, is we need to factorize the given function. So we need to take F of X, and we need to say that it's equal to X2 + 3 X minus 10. Divided by X plus 5, and when we factorize, we will say that this is equal to X + 5 multiplied by X minus 2 divided by X + 5. And when we cancel out our like terms, which in this case it's X + 5, we will find that it's equal to X minus 2. And we need to note, we need to make a note, this is very important that X cannot equal 5, and we need to avoid division by 0 in the original function, and because we need to avoid division by 0 in the original function, X cannot equal 5. So now we can begin by finding the first derivative, which we'll call this F. Of X and F of X is equal to D by DX because we're taking the derivative with respect to X. So F of X is equal to D by DX of X minus 2. So that will mean that our first derivative F of X will be equal to 1, and that is our first answer. Hooray, we did it. Now we could solve for our second answer to find the second derivative, which we'll call this F of X and F of X will be equal to D by DX of. F of X. So that will mean that We will take, when we take our second derivative, that will mean that F of X will be equal to D by DX 1. Which will mean that as a result, F of X, our second derivative will be equal to 0. So F of X is equal to 0, and that is our second derivative. Hooray, we did it, and just to make sure we could write that this is the 1st. Second, so now we can solve for our third derivative. So our third derivative, we'll call this F of X and F of X will be equal to D by DX of F of X. Fantastic. So that will mean that when we take F of X is equal to D by DX of 0. That will mean that F of X is equal to 0, and that is our final answer for our third derivative. Hooray, we did it. We found all three answers. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Find f′(x), f′′(x), and f′′′(x) for the following functions.
f(x) = (x² - 7x - 8) / (x + 1)

Key Concepts

Differentiation

Higher-Order Derivatives

Quotient Rule

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