Find and simplify the derivative of the following functions.

Question

h(x) = (x − 1)(x³+ x² + x+1)

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with derivatives. This problem says to determine the derivative of the function p of X, which is equal to the quantity of 3 X minus 2 in quantity multiplied by the quantity of X cubed plus 2 X squad minus X minus 2. We're given 4 possible choices as our answers. For choice A, we have P X is equal to 12 x cubed plus 12 x quad minus 18 x plus 9. For choice B, we have P X is equal to 12 x cubed plus 12 x quad minus 14 X minus 4. And for choice C, we have P X is equal to 9 X2 + 4 X minus 3. And for choice D we have P X is equal to 9 x 2 plus 11 x minus 7. Now, we're asked to take a derivative of our function PF X. In order to take this derivative, we're going to need to use our product rule. So, we call for our product rule that we're going to have the derivative with respect to X. Of the quantity of F of X multiplied by G of X. I want to take a derivative of this product, this is going to be equal to F of X. Multiplied by G of X. Plus G of X. Multiplied by F of X. So we're gonna use our product rule to evaluate this derivative. That means that P of X. is going to be equal to and here applying our product rule, we'll have our first term in our product, which is the quantity of 3 X minus 2 in quantity, that gets multiplied by the derivative with respect to X of our second term, which is the quantity of X cubed. Plus 2 X squared. Minus X minus 2 in quantity. Then we'll have plus our second term, which is the quantity of X cubed. Plus 2 X squad minus X minus 2 in quantity, and that gets multiplied by the derivative with respect to X of the first term in our product, which is the quantity of 3 X minus 2. So now taking the derivative, this this is going to be equal to the quantity of 3 X minus 2. Multiplying the quantity. And when I take the derivative with respect to X of Xecute, I get 3 X2. Plus, the derivative of 2 X2 gives me 4 X. Then we'll have minus the derivative of X is just 1, and then when I take the derivative of -2, that gives me 0. So in the quantity after -1. Then we will have plus. The quantity of X cubed. Plus 2 X squad. Minus X minus 2 in quantity, multiplied by the derivative spec X of the quantity of 3X minus 2, and so, taking that derivative, we're just going to get 3, since the derivative of 3 X is 3, and the derivative of -2 is 0. So performing the multiplication now, this is going to be equal to. 9 X cued Plus 12 X squared. Minus 3 X Minus 6 X squared. -8 X. Plus 2 Plus, 3 execute. Plus 6 X squared. -3 X. -6. So now all we need to do is collect like terms. So this is going to be equal to. If I look at my cubic terms, I have 9 X cubed plus 3X cubed, that gives me 12 x cubed. Then for the square terms, I'll have 12 X2 minus 6 X2, plus 6 X2, that gives me plus 12 X2. For our linear terms in X, we'll have -3 X 8X minus 3X that gives me 14 X. And for our constant terms, I'll have + 2 minus 6, which will give me -4. And so this is the answer to the question that we are looking for, and when I compare our answer here to our answer choices, the correct answer choice corresponds to answer choice B. So I hope that this explanation has been useful, and I'll see you in the next video.

Find and simplify the derivative of the following functions.
h(x) = (x − 1)(x³+ x² + x+1)

Key Concepts

Derivative

Product Rule

Simplification of Derivatives

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