13-26 Implicit differentiation Carry out the following steps.

Question

a. Use implicit differentiation to find dy/dx.

³√x+³√y⁴ = 2;(1,1)

Accepted Answer

Welcome back, everyone. Given the equation, the 4th root of X plus the 4th root of Y 5th equals 1, use implicit differentiation to find the YD X. A says it's -1 divided by 5 x 3/4 Y B 1 divided by 5 X 3414. C-1 divided by X 3/4 Y 1/4 and D 1 divided by X 3/4 Y 14. Now, if we're going to implicitly differentiate our function, our equation, then let's try to write it in a way that makes it easier to differentiate, easier to read, OK? Now, let's take our term. The 4th root of X plus the 4th root of Y to the 5th equals 1. And let's use what we know. Let's rewrite our our roots as powers, OK? So when we do that, it's gonna be X to the power of 1/4 plus Y to the power of 5/4 equals 1. No, it might be, it might seem easier to differentiate because we will be able to apply the poor and the chain rules. So differentiating, OK, we're gonna differentiate, so by differentiating. Then what that really means is that for these three separate terms we're going to find the derivative with respect to X of X to the 4th. Plus the derivative with respect to x of Y 5/4, OK. And the derivative with respect to X of 1. Now what wilt thou do be. Well, using the poor rule to differentiate X to the 4th, we're gonna bring the power down 1/4 and subtract 1 from the poor. So that's 1/4 minus 1, which is gonna be negative 3/4. So that's what's gonna happen for our ex term. Now, for the Y term, we can use the chain rule, which basically will bring down our power 5/4, OK. Uh, subtract 1 from the power of Y, uh and 5/4 minus 1 equals 1/4, and then differentiatey with respect to X. We're gonna get the YD X since Y is a function of X here, OK? And no, on the right hand side, the derivative of 1 is 0 because 1 is a constant. Now that we have it in this way, let's go ahead and see if we can isolate or we can solve for DUIDX. If we do some rearranging here, OK, then we should get 5/4 of Y to the 4th multiplied by DYD X. To be equal to -14 multiplied by X to the negative 34, OK? Now here, notice we can divide both sides to get D Y D X we can divide both sides by 5/4 of Y to the 4th, and when we do that. Then we should get DD X to be equal to 4th of X 4 multiplied by X 3/4 divided by 5/4 multiplied by Y 4th. And now here we can simplify, we can cancel out our fourths to get negative X to the negative 3/4. Let me write that properly here divided by 5 y to the fourth, and we can even write that further. I can take a step further to write x in the denominator using our rules of courses. When we do that, then we're gonna get -1 divided by 5 X to the 3/4 multiplied by y to the 4th. Therefore, this is going to be the do, let me write a negative sign better here. This is going to be the derivative of Y with respect to X for our function. If we look back at our answer choices, that tells us A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

13-26 Implicit differentiation Carry out the following steps.a. Use implicit differentiation to find dy/dx.³√x+³√y⁴ = 2;(1,1)

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13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
³√x+³√y⁴ = 2;(1,1)