13-26 Implicit differentiation Carry out the following steps.

Question

b. Find the slope of the curve at the given point.

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 to the 5th equals 2, use implicit differentiation to find the slope of the tangent line at the 0.11. A says it's 1/5, B1, C -1, and D says it's -15. No. For us to find a slope of the tangent line at the 0.11 then we need the derivative of our equation with D Y D X, the derivative of y with respect to X. At the 0.11. But before we can find it at the 011, which means that we're just going to substitute X X41 and Y values for 1, we'll need to find DYD X. So let's differentiate using implicit differentiation, and then we can plug X and Y values of 1 into our derivative. Now to differentiate it implicitly, we're going to differentiate the left hand side with respect to X and the right hand side with respect to X. In other words, OK. What it really means that we're doing is that we're going to find a derivative with respect to X of the fourth root of X, which we can rewrite as X to the 4th. The derivative with respect to X of the fourth root of Y to the 5th, which we can rewrite as y to the pore of 5/4s. And the derivative with respect to X of 2. Now what will those derivatives be? Well, we can differentiate X to the 4th using the poor rule, and that's gonna give us a 4th of X to the power of -3/4. We bring down the power and then subtract 1. Next, we can differentiate the y term using the chain rule, treating Y as the inner function. So we can bring down our po 5/4, subtract one from the power of our al function, so that's y 5/4 minus 1, which is going to be Y 4th, and then we can write the derivative of Y, which is D YD X, the derivative of Y with respect to X. And now on our right hand side, the derivative of 2 with respect to X will be 0 because 2 is a constant. Now that we have this as our derivative, we can solve for a DYDX, OK? And if I make some space down here, that means we can rewrite this as 5/4 of Y 4 multiplied by uh DUIDX. OK. Equal to 14th multiplied by x 3/4 and know if we divide both sides by 5/4 of Y and factor out of fourth from both terms, then we'll get the UID X. To be equal to x to the 3/4 divided by 5 y to the 4th. Now that we have the derivative of y with respect to X, we can go ahead and serve, get the derivative at 11 to get the slope. So now at 11, you are the X. Is going to be equal to -1 to 3/4 divided by 5 multiplied by 14th. Now we know that 1 race to anything is equal to 1. So this is going to simplify the -15. Thus the slope at 11 is 1/5. If we look back at our answer choices, that tells us that D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

13-26 Implicit differentiation Carry out the following steps.b. Find the slope of the curve at the given point.³√x+³√y⁴ = 2;(1,1)

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13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.
³√x+³√y⁴ = 2;(1,1)