51–56. Second derivatives Find d²y/dx².
2x²+y² = 4

Question

51–56. Second derivatives Find d²y/dx².

2x²+y² = 4

Accepted Answer

Welcome back, everyone. Calculate the second derivative of 5 X cubed plus 9 Y cubed equals 19. For this problem, we're going to apply implicit differentiation. So let's rewrite the given equation. It says 5 Y cubed plus 9 Y cubed equals 19. So we take the derivative of the left hand side and we equate it to the derivative of the right hand side. So now for the first term we can factor out the constant. We can say that it is 5, and then we differentiate X cubed. Let's apply the power rule. We have to recall that we bring down the exponent, so that's 3. We then write X and we decrease the original exponent by 1 unit. This gives us 3 X2, and then we're going to add 9, which is our constant. Apply the power rule for 34 Y cubed, which is 3 Y squad, but now Y is a function of X. So this means that we have to apply the chain rule and multiply by Y, which is the first derivative. And on the right hand side, we have the derivative of 19, that is a constant, so the derivative of 19 is 0. Now, let's simplify. We're going to get 15 X squared. Plus 27. Y 2 Y equals 0 and now let's solve for Y. So now Y is going to be equal to. We're going to move 15 x 2 to the right, which gives us negative 15 x 2 and divide by the factor in front of Y, which is 27 Y2. Now let's simplify. Let's understand that 15 and 27 are both divisible by 3, right? So we're going to get -5 X 2 divided by 9 Y2. That's our first derivative. Now, to identify the second derivative, we simply want to differentiate the first derivative. So why prime. Is the derivative of negative. 5 divided by 9 we can factor out the constant right so -5 divided by 9. Derivative of X divided by Y2d. We can also apply the rules of exponents, right, because this is -5 divided by 9. X divided by Y squared. So now, let's evaluate the derivative. Before that, we're just going to include the prime sign to clearly show that we're evaluating the derivative. And here, we will take -5 divided by 9. So that's our constant. And as we can see, the outer function is an exponent. So once again, we are applying the power rule. We bring down the two. So we multiply by 2. Which is multiplied by x divided by Y raises the power of 2 minus 1, which gives us an exponent of 1. But now once again we have an inner function, so we have to multiply by the derivative of X divided by y, right? Well done. What we have to do from here is evaluate the derivative of X divided by Y using the quotient rule. So x divided by y derivative is equal to the derivative of the top function. So X multiplied by the bottom function Y minus the top function X multiplied by the derivative of the bottom function, which is Y. And then divided by the square of the bottom function we get Y squared. And now we're going to substitute this into the expression, right? So, first of all, we have to understand that we get negative 10 divided by 9. Multiplied by X. Divided by Y. Which is then multiplied by our derivative. The derivative of X is 1, so this means that we're going to get a Y. Minus x multiplied by the first derivative. So x multiplied by. -5 divided by 9. X divided by Y2. So here we are performing substitution, right? And all of that is going to be divided by Y2d. Now, we have a lot to simplify, so let's just continue, right? Because we are done with evaluating all of the derivatives, we just need to simplify the final answer. First of all, we're going to take our constant, which is -10 divided by 9. And we're going to use a single numerator and a single denominator. So in the numerator, we're going to get -10. X In parent, we can see that we're multiplying by y. Minus and because they're saying negative term, we can just replace that negative sign with a positive sign. And then This is X multiplied by 5 9s, multiplied by X2 divided by Y2. So what does that mean? Well, X multiplied by X2 gives us X cubed, so we're going to get 5. Nice. X cubed divided by Y2. And now in the denominator, we have Y multiplied by Y2, so this gives us Y cubed. And now from here we can find the common denominator, right? So, essentially, considering our terms, what we can do is simply We write Our sum as follows, we're going to take out 10 X. And our imper see. Our Y is going to become Y cubed. We're going to add. 5 x cubed. And I notice that it's 5 9s, right? So we actually have 9 Y 2d and this means that our common denominator is going to be 9Y2. Which means that Y is not only multiplied by Y2 but also by 9. So we are going to get 9 Y cubed plus 5 X cubed divided by 9 Y quad. And then all of that is divided by 9 Y. cue. From here, we can understand that we can rewrite those two denominators as a product. On one single denominator because we have a double division, right, we can take -10 X and we can distribute relative to the numerator. So if we just factor out the negative sign, we can consider 10X as a positive value multiplied by 9 Y cubed, and this gives us 90 X Y cubed. And then we have 5 X cubed multiplied by 10 X. So that's plus. 50 X to the power of 4th, and then combining our two denominators, 9 multiplied by 9 gives us 81 and Y2 multiplied by Y cubed gives us Y is the power of fifth. Which is the same thing as negative 50 x to the power of 4th plus 90 x y cubed divided by 81 y to the power of fifth. And this would be our 2nd derivative, so we can essentially label it as the 2nd derivative for our answer. Well then we have our final answer. Let's label it and thank you for watching.

51–56. Second derivatives Find d²y/dx².2x²+y² = 4

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51–56. Second derivatives Find d²y/dx².
2x²+y² = 4