27–76. Calculate the derivative of the following functions.

Question

$y = {(x^{2} + 2 x + 7)}^{8}$

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with derivatives. This problem says to determine the derivative of Y is equal to the quantity of 2 X2 + 3 x + 4 in quantity rates to the 4th. We're given 4 possible choices as our answers. For choice A, we have the derivative of Y with respect to X is equal to 4, multiplying the quantity of 2 X2 + 3 x + 4 in quantity cubed, multiplied by the quantity of 4 X + 3. For choice B, we have the derivative of Y with respect to X, and that's going to be equal to 4, multiplying the quantity of 2 X2 + 3 x + 4 in quantity cubed. For choice C, we have the derivative of Y with respect X is equal to 4 multiplied by the quantity of 2x2 + 3 x + 4 in quantity cubed multiplied by 2 x + 3. And for choice D, we have the derivative of Y with respect to X is equal to 4, multiplying the quantity of 4 X plus 3 in quantity cued. Now, to find the derivative of Y with respect to X, we're actually going to need to use the chain rule. So we call for the chain rule that we have the derivative of Y with respect to X is going to be equal to the derivative of Y with respect to U, multiplied by the derivative of U with respect to X. So, the first thing we need to do is identify our inner function, U, and for this we're going to use the quantity that is inside the parentheses. So you, Going to be equal to 2 X2 + 3 X. Plus 4. Then we can write Y, which is now going to be a function of you. As you race to the 4th. So now I have these two expressions, we can use them to calculate our derivatives that go into our chain rule. So, the first one thing we're going to calculate is our derivative of U with respect to X. And this is going to be equal to the derivative with respect to X of the quantity of 2 X2 + 3 X + 4. And so here to take this derivative, we call your power rule for derivatives, that is the derivative with respect to X of the quantity of X to the N, is going to be equal to N multiplied by X raised to the N minus 1. And we're also going to need to take a derivative of a constant. So remember that rule, that is the derivative with respect to X of our constant C, it's going to be equal to 0. So we use these two rules to calculate our derivative, and this is going to be equal to. 4 X Plus 3. Since for 2 x 2, I'll be multiplying the 2 in front by the power 2 in the exponent which will give me 4, and the subtracting 1 from 2, which will give me the power of 1, which is just X. And then for the 3 X term, we'll be taking a derivative of X-rays to the 1, so 3 multiply 1 by 1 is 3, and 1 minus 1 is 0. So X to 0 is just 1, so the 3 just becomes a constant term. Now we do the same thing for our derivative. Of Y with respect to you. This is going to be equal to the derivative with respect to you, of our quantity of you raised to the 4th. And so again, applying our power rule or derivatives, this is going to be equal to. Or multiplied by you cut. So now we can take these two quantities and substitute them back up into our chain rule formula. We'll have the derivative of Y, with respect to X is equal to or our derivative of Y with respect to U. We're going to have 4 multiplied by u cubed. However, for you, we want to substitute back in the 2 X2 + 3 X + 4, since we want our final answer as a function of X. So we'll have 4 multiplied by the quantity of 2 X 2 plus 3X. Plus 4 in quantity cubed. And then this is going to get multiplied by our derivative of view with respect to X, which is the quantity of 4 X plus 3. And so this is the answer that we are looking for, and if we compare our answer to our answer choices, the correct answer choice turns out to be answer choice A. So I hope that this explanation has been useful, and I'll see you in the next video.

27–76. Calculate the derivative of the following functions.
$y = {(x^{2} + 2 x + 7)}^{8}$

Key Concepts

Derivative

Chain Rule

Power Rule

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