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

Question

$y = \sqrt{10 x + 1}$

Accepted Answer

Hi everyone. Let's take a look at this practice problem dealing with derivatives. This problem says to determine the derivative of Y is equal to the square root of the quantity of 3X2 + 2. We give 4 possible choices as our answers. For choice A, we have the derivative of Y with respect to X is equal to 3 x divided by the square root of the quantity of 3 x 2 + 2. For choice B, we have the derivative of Y. With respect to X is equal to 3 x divided by the quantity of 2 multiplied by the square root of the quantity of 3 X2 + 2. For choice C, we have the derivative of Y with respect to X is equal to 1 divided by the quantity of 2 multiplied by the square root of the quantity of 3 x 2 + 2. And for choice D we have the derivative of Y with respect to X is equal to 1 divided by the square root of the quantity of 3 x 2 + 2. Now, in order to take the derivative of this function, we're going to need to use our chain rule. So recall for the chain rule that the derivative of Y with respect to X, it's going to be equal to the derivative of Y with respect to U, multiplied by the derivative of U with respect to X. Where you here is going to be our inner function. So, to identify our inner function, we're going to look at the uh equation that we're given in the problem, and we're going to take our function U to be what's underneath the radical. So we'll have U equal to. 3 X2 + 2. So that's going to be our inner function. So that means that we can write Y as a function of you, which is going to be our outer function, and that's going to be equal to you raised to the 12 power. Since taking the square root of a quantity is the same as raising that quantity to the 12 power. So now we can use our inner and outer functions to calculate our derivatives in the chain rule. So, the first derivative that we're going to calculate is the 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 3 X2 + 2. So here we need to recall our power rule for derivatives, that is that the derivative with respect to X, of the quantity of X raised to the N is equal to N multiplied by X raised to the N minus 1. And we also need to recall our um relationship for taking derivatives of constants. So the derivative with respect to X of a constant C is equal to 0. So we can use these two rules to calculate our derivative of you with respect to X. So, applying those two rules, our derivative is going to be equal to. Here, we'll have 3 X2, and so I'll multiply the 3 by the 2, which will give me 6, and then I'll subtract 1 from the 2 in the exponent, which will give me X to 1, which is just X. And then we'll be taking a derivative of 2, which is a constant, which will give me zero, so I won't even include that term in our expression. So, our entire derivative is just going to be 6 X. We also need to calculate the derivative of Y with respect to you. And this is going to be equal to the derivative with respect to you. Of the quantity of you raised to the one half power. And so again, applying our power rule. Or derivatives, this is going to be equal to 1/2 multiplied by U raised to the minus 1/2 power. So we now take these two expressions and substitute them back into our formula for the chain rule. And so we'll have the derivative of Y with respect to X, it's going to be equal to. 1/2 multiplied by you raised to the minus 1/2. But we want our final answer as a function of X. So, for you, we're going to substitute back in 3X2 + 2. So, I'll have 1/2 multiplying the quantity of 3 X2. Plus 2 In quantity, raised to the minus 1 half hour, and then that's going to get it multiplied by our derivative of you with respect to X, which in this case is just 6 X. And so we can actually simplify this expression. So we'll have this equal to. I'll have the 6 multiplied by the 1/2, which will give me 3 X since we also have an X here that I'm going be multiplying by. And that is going to be divided by the square root of the quantity of 3 X2. Plus 2. So here I've written the quantity of 3 X2 + 2 in quantity rates to the minus 1/2 as 1 divided by the square root of the quantity of 3X2 + 2. So this is the answer that I was actually looking for. And if I compare this to my 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.
$y = \sqrt{10 x + 1}$

Key Concepts

Derivative

Chain Rule

Square Root Function

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