If x¹/³ + y¹/³ = 4, find d²y/dx² at the point (8, 8).

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Accepted Answer

Welcome back, everyone. Given the equation 2 X 4 plus 3 Y equals 10, find the second derivative at the 0.16 16. A says it's 5 divided by 192, B 1 divided by 32, C 1 divided by 24, and the D 5 divided by 96. No. If we're going to find the second derivative of our equation, it just means that we need to differentiate it twice, and then to find it at the 0.16 16, we plug 1616 into our second derivative. So let's start by finding this first derivative. Now that means we're going to differentiate. Our equation with respect to X, OK. And no notice that in our equation we have Y on the left hand side, so that means we'll need to implicitly differentiate. No, the second, no, sorry, the derivative first to differentiate 2 X to the 4th, this derivative will be equal to a half of X to the negative 3/4, OK? Next, the derivative of 34 is going to be 3/4 of. Why to the 3/4. DIX And then the derivative of 10 is 0 because 10 is a constant. No, we can rearrange the so for DUIDX. If we do that, OK, if we subtract the 1/2 of X to the negative 3/4 from both sides, then we should get 3/4 of Y to the negative 3/4 DD X. To be equal to. Negative half of X to the negative 3/4, OK? I know how to get DYDX we can multiply by the reciprocal of the term that's multiplying it. In other words, we can multiply our entire equation by 4/3 of Y to the core of 3/4, OK? And no, when we do that. Let me come over to the right. Then we should get DUIDX. To be equal to negative 2/3 multiplied by Y 3/4 divided by X of 3/4. thus this is our first derivative. To get a second derivative, we'll differentiate DYDX again with respect to X. So by doing that, differentiating the ID X with respect to X. OK Now if we take a good look at DYDX, it is a quotient, and we can differentiate quotients using the quotient rule of differentiation. Recall that rule basically tells us that if we have a function Y that's equal to a quotient U divided by V where U and V are functions of X, then the derivative of Y is going to be equal to Vu minus UV divided by V2. Where U and V are the derivatives of U and V respectively. To know In our problem, if we let you be equal to our numeratory to the power of 3/4, OK, then the derivative of you would be equal to 3/4 of Y to the negative 4 D Y D X, OK. Likewise, if we let V our denominator, the X 3/4, then the derivative of V is going to be equal to 3/4 of X to the negative. Know that we have uh our our derivatives here and if we treat negative 2/3 in our first derivative as a constant, then that means the second derivative. It's gonna be equal to -2 1/3. Multiplied by Vu, OK. So that's going to be X 34, multiplied by 3/4 of Y to the negative 4 DD X. OK. And we're gonna subtract the UV. So that's minus Y 3/4 multiplied by 3/4 of X to the negative, OK? And now we're dividing all of this by V2. So that's X of 3/4 squared. No, let's try to simplify. If we think about it here, we know what DUIDX is, so we can replace that in our first expression. So now we can write 2/3 multiplied by X 3/4 multiplied by 3/4, OK, of the negative 4 multiplied by 2/3 of Y 3/4 divided by X 34. Remember this is the the X. No, on for a second term, you can rewrite that as minus 3/4 of Y 3/4 X to the negative 4s. I know we can divide through in our entire equation, OK, by X to the power of 3 halves. In this case, notice that x 3/4 cancers X 3/4, Y the negative 4 multiplied by Y to the power of 3/4 because both of them are the same base, then we can add their powers and the 3/4 multiplied by 2/3 would leave us with negative half, OK. So our first term is gonna now look a bit different. So this is now gonna be negative 2/3, OK. Multipplied. By negative half multiplied by Y to the power of a half, OK. Minus 3/4 of Y 3/4 multiplied by X 4. Divided by, OK, let me just clean that up here. Divided by X to the power of 3 halves, OK. Now, could we, could we simplify this any further? Well, at this point, it would probably be a good idea for us to try to, uh, for us to try to. Substitute our points. Now that we have our second derivative, OK. So at the point, this is our second derivative, and now at 1616. That is where X is 16 and Y is 16. Then our second derivative. OK, let me put that in black here. Our second derivative. It's gonna be equal to -23. Multiplied by -5, multiplied by 16 to the power of a half, OK, which is, well, that's basically um that's basically 4. So it's negative 5 multiplied by 4 minus 3/4 of 16 of 3/4, which would be 8, OK. Multiplied by X 25. 16 to the negative quarter would be a half, OK? I know all of this is being divided by X 16 to the power of 3 halves, which would be 64. Now when we go ahead and simplify here, we're gonna get 2/3, multiplied by -5 divided by 64, -5 would be the result of everything that's happening in our numerator here. A negative 2/3 multiplied by -5 divided by 64 is going to be equal to 5 divided by 96. Thus, this is the value of the second derivative at the 0.1616. 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.

If x¹/³ + y¹/³ = 4, find d²y/dx² at the point (8, 8).

Key Concepts

Implicit Differentiation

Second Derivative

Chain Rule

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