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

Question

$\sqrt[3]{x^{2} + 9}$

Accepted Answer

Welcome back, everyone. In this problem, we want to determine the derivative of Y equals the 4th root of X2 + 11. A says it's x divided by 2 multiplied by the 4th root of X + 11. B says it's x divided by 4 multiplied by the 4th root of X2 + 11 cubed. C says it's x divided by 2 multiplied by the 4th root of X2 + 11 cubed, and the D says it's 1 divided by 4 multiplied by the 4th root of X2 + 11 cubed. Now, let's take a good look at our function Y equals the 4th root of X + 11. Notice that this is a composite function. It has an inner function, X2 + 11, and an alter function, the 4th root of that value. So because it's a composite function, we can apply the chain rule of differentiation. Recall, the chain rule tells us that if we have a function Y equals U to the power of N where U is a function of X. Then the derivative of y with respect to x equals the derivative of y with respect to u multiplied by the derivative of u with respect to X. So if we let you be equal to our inner function X2 + 11, if we differentiate our outer function Y with respect to you and differentiate our inner function with respect to X, we can find their product to get the derivative of Y with respect to X. So let's break that down. Let's go ahead and do that. Starting with our auto function, Y is going to be equal to the 4th root of U, OK, which we can rewrite as u to the 4th. Now if we differentiate why with respect to you, OK, then. I should have written that in red, but it's OK. I'll keep it in the same color. DYDU, the derivative of why with respect to you. To find it, we're going to bring down the power based on the poor rule and then subtract one from the poor. So that's gonna be you raised to the poor of 4 minus 1, which equals -3/4. Now if we write back our function U here using what we know about. Let me just make a note here. What we know about powers, we could rewrite this as 1 divided by 4 you raised to the power of 3/4, and then we know. That you we already know that U is equal to X2 + 11. So this is really going to be then 1 divided by 4 multiplied by X2 + 11 raise to the power of 3/4. Now again, using what we know about uh about powers here, recall, let's put a star here that based on fractional fractional law for powers. If we have a power, or if we have a base, sorry, raised to a fractional power a divided by B, then we can also write it as the root or the root, sorry, the B root, I know B sounds weird, but be root of X to the power of A, OK. So in that case, if we apply that rule, we write the denominator as the root and the numerator as the power, then we could write D YDU as one divided by 4 multiplied by the 4th root of X2 + 11 cubed. So this is going to be DIDU. Now, what about derivative of you with respect to X? Well, since you equals x2 + 11. To find DUD X, we differentiate X squad using the power rule, which would give us 2 X and we differentiate 11 because it's a constant, it would be equal to 0. Thus, DUD X are the derivative of you with respect to X equals 2 X. Let me just bring it a bit closer here. So know that we have DUDX. And we have DUIDU. We can apply the chain rule. Because no, that means DUID X then is going to be equal to 1 divided by 4 multiplied by the 4th root, OK, of X2 + 11 cubed. Multiplied by DUDX which is 2 X. Notice here we can cancel 2 from the numerator and the denominator, sorry, 2 goes into 42 times. So we could write this then as x divided by 2 multiplied by the 4th root of X + 11 cubed. This is DUIDX, therefore. C is the correct answer. Thanks for watching everyone. I hope this video helped.

27–76. Calculate the derivative of the following functions.
$\sqrt[3]{x^{2} + 9}$

Key Concepts

Derivative

Chain Rule

Power Rule

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