Find and simplify the derivative of the following functions.

Question

f(x) = e^x(x³ − 3x² + 6x − 6)

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with derivatives. This prompt us to find the derivative of the function G of X, which is equal to E X, multiplied by the quantity of 5 X cubed minus 15 x 2, plus 30 X minus 30. We're given 4 possible choices as our answers. For choice A, we have G of X is equal to 30x2 multiplied by E raised to the quantity of 2 X. For choice B, we have G of X, which is equal to 10 x cubed, multiplied by E to the X. For choice C we have G of X is equal to 15X2 multiplied by E raised to the quantity of 2 X. And for choice D we have G of X, which is equal to 5 X cubed multiplied by E to the X. Now we're asked to find the derivative of our function G of X, and in order to take this derivative, we're going to need to use our product rule. To recall for the product rule that we have the derivative with respect to X of the quantity of F of X multiplied by G of X. And 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 going to apply our product rule here to our function G of X. So we're going to be calculating G of X. And this is going to be equal to. And if we look at our protocol, we're gonna have the first um term of our product, which is going to be EF X, E, Ease to the X. And that's gonna be multiplied by the derivative with respect to X of the quantity. Of our second term, which is 5 X cubed minus 15 X squared. Plus 30 X. -30. In quantity. And then we'll have plus, the second term on our product, which is going to be the quantity of 5 X cubed minus 15 X squared. Plus 30 X. Minus 30 in quantity, and that gets multiplied by the derivative of our first term, which is the derivative with respect to X. Of the rates to the X. So now we need to do is evaluate our derivatives in these products. So this is going to be equal to. We'll have Ease to the X, multiplied by the quantity, and when we take a derivative of 5 X cubed, we get 15 X2, then we'll have minus, we'll take the derivative of 15X2 which will give us 30 X. Then we'll have plus the derivative of 30 X is 30. And then when we take the derivative of -30, with respect to X, that gives us 0. So we'll end our quantity after the plus 30. Then we'll have less. The quantity of 5 X cubed minus 15 X squad. Plus 30 X. -30 In quantity, and that gets multiplied by the derivative with respect to X of E to the X, which is E rate to the X. Now, I have an E to the X in both of these terms, so I can factor it out. So this is going to be equal to E race to the X. Multiplied by the quantity, and now I can just combined like terms for everything that's left. So, looking at our cubic terms, we'll just have 5 multiplied by X rates to the 3 power. Or our quadratic terms will have 15 x2 minus 15 x 2, so that adds up to 0, or our linear terms will have minus 30 X plus 30 X, which add to zero, and finally, for our constant term, we'll have 30 minus 30, and so those add up to 0. So we just really have E to the X multiplied by 5 X cubed. So just rearranging those two products, we're going to have 5 X cubed. Multiplied by E rates to the X. So this is the answer that we are looking for, and if I compare that to my answer choices, the correct answer choice corresponds to answer choice D. 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.
f(x) = e^x(x³ − 3x² + 6x − 6)

Key Concepts

Derivative

Product Rule

Exponential Function

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