51–56. Second derivatives Find d²y/dx².
x⁴+y⁴ = 64

Question

51–56. Second derivatives Find d²y/dx².

x⁴+y⁴ = 64

Accepted Answer

Welcome back, everyone. Calculate the second derivative of X to the power of 6 plus y to the power of 6 equals 121. For this problem we're going to apply the concept of implicit differentiations. We have X, the power of 6, plus y is the power of 6. On the left hand side, we're going to take the derivative and we're going to set it equal to the derivative of 121. So first of all, we're solving for the first derivative and then we will proceed with the 2nd 1. For the term on the left, let's apply the power rule. The derivative of X the power of 6 is 6x the power of fifth. That's power rule. For y to the power of 6, we're going to get 6 to the power of 5th, but Y is a function of X, so we also need to multiply by Y. Right? Based on the chain rule, we get the first derivative, and we're going to set it equal to 121 derivative, which is 0 because this is the derivative of a constant. And now what we can do is simply solve for Y, right? So if we move 6 X the power of 5th to the right, we're going to get negative 6 X to the power of fifth. And then we have to divide by the leading term, so that's 6 to the power of fifth. And this allows us to isolate Y and for simplicity, well, we can notice that 6 and 6 can be canceled out and we can apply the properties of exponents to write it in the form of negative X divided by Y all raised to the power of fifth. Now what can we do from here? Well, essentially we have the first derivative. Now our goal is to evaluate the second derivative, which means that we are evaluating the derivative of negative X divided by Y raised to the power of fifth, and we are basically factoring out the negative sign, right? It makes no difference. So we take negative. And now let's understand that this is a composite function. So first of all, we have to apply the power rules, so we bring down the 5. So this gives us -5 multiplied by X divided by Y, raise the power of 5 minus 1, which is 4. So here we have applied the power rule and now based on the chain rule, we have to multiply the result by the derivative of X divided by Y. And this is where we need to apply the quotient rule. So let's recall that we first will take the derivative of the top function. So that's x multiplied by the bottom function, which is Y. Subtract the first function multiplied by the derivative of the bottom function. So we subtract xy and then divide by the square of the bottom function. And if we use this result overall we're going to get -5. X divided by Y is the power of 4th multiplied by. What is the derivative of X well based on the Our rule, this is 1, so we get Y minus X Y. And here we have to make use of Y. We know that it is equal to negative, x divided by Y is the power of fifth. So if we have 2 negative signs, we can just replace it with a positive sign. And x is now multiplied by. X divided by y raises the power of fifth. And now in the denominator we have Y2d. Now, we have the expression of the second derivative. All that we have to do is try to simplify it. So we're going to distribute. The 4th power for the first term, this gives us -5. And now let's use the common numerator and denominator. In the numerator, we are going to have X the power of 4th. What else do we have? Well, in prints, we're going to get Y. Plus X multiplied by x to the power of 5th, so this gives us X to the power of 6th. Divided by y to the power of fifth. Now let's consider the denominator. Or the denominator. We're going to have Y 4 multiplied by Y2. So this is the power of 6. And now, as we can see in the numerator, we have an additional denominator. So let's identify the common denominator once again. We have -5 in front. Then we're going to have X to the power of force multiplied by y if we distribute, right? And if we want to use a common denominator, which is why it's the power of 5th. Well, essentially we have to multiply that by the power of 5th, which gives us y to the power of 6th in the. Numerator and then we are adding X to the power of 6th, and all of that is divided by Y to the power of 6th. And because we have a double division, well, simply speaking, what we can do is combines the power of fifth with whites the power of 6th and multiply them. So this gives us whites the power of 11th and now we can multiply -5 by each term in the numerator. First of all, let's leave the negative sign in front. And we get 5 X to the 4th, Y to the 6th, plus. 5 X to the power of. Now x the power of 6th actually becomes x to the power of 10th because we have x the power of 4th multiplied by y. This gives us X the power of 4thy to the power of 6 and also X the power of 4th is multiplied by x the power of 6, right? So we have to clear that mistake and replace X the power of 6 with X to the power of 10th. So the term is actually x the power of itself. And that's it. Overall, our result is negative. 5 x is the power of 4th, white's the power of 6th, plus 5 X is the power of 10th, all divided by white to the power of 11th. Let's label it as our final answer for the 2nd derivative. Thank you for watching.

51–56. Second derivatives Find d²y/dx².x⁴+y⁴ = 64

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51–56. Second derivatives Find d²y/dx².
x⁴+y⁴ = 64