Finding The Second Derivative by Implicit Differentiation

Number 54. Which of the following is equal to d^2 y over dx^2? So this represents the second derivative of y with respect to x. That's what we need to find. So we got to find it using implicit differentiation. So let's differentiate this equation with respect to x. The derivative of x cub is 3x^2. The derivative of y cub is 3 y^2 * dydx. And the derivative of 35 is zero. So now we need to solve for dydx. Let's subtract both sides by 3x^2. So we're going to have 3 y^2 dydx is equal to -3x^2. Our next step is to divide both sides by 3 y^2. These will cancel and we can also cancel three. And so we're going to get this dydx is equal to -x^2 over y^2. [snorts] So now what we're going to do is we're going to differentiate this equation with respect to x and differentiate the left side with respect to x. When we do that the derivative of dydx with respect to x is d^2 y over dx^2. And you can kind of see that dx * dx is basically dx^ 2 and d * dy is like d^2 y. Over here we need to use the quotient rule. So f / g the derivative of that is going to be g f prime minus f g prime over g^2. So we can see that f is the numeratorx^2 g is y^2 fprime is going to be -2x g prime will be 2y * dy / dx. So let's start over here. G FP prime G is Y^2 FP prime is -2X and then minus F. F is -x^2 * G prime. That's going to be 2y * dy dx. And this is going to be all over g^2. So that's y^2 squared. So multiplying these two we're going to get -2x y^2. Here we could cancel the two negative signs. So it's going to be plus 2x^2 y and then * dydx. On the bottom we have y^2 * 2 is 4. So we get y 4th power. Now notice that dydx is equal to -x^2 over y^2. So we want to replace dy dx with that. So this is going to equal -2x y^2 + 2x^2 y * -x^2 / y^2 So this y^2 right here, it's equivalent to y * y. And notice we could cancel one of the y's. So what we now have is -2x y^2. Here we have 2x^2 *x^2.

So that's -2 x to the 4th. And we have a y on the bottom. And this is all divided by y 4th. Now what I'm going to do in the next step, I'm going to multiply the numerator and the denominator by y. So this will give me -2x y^ the 3 power. Here the y variables will cancel. So - 2x 4th and y 4th * y first power. We need to add the exponents. So 4 + 1 is 5. Now, what I'm going to do at this point is I'm going to factor the GCF in the numerator. So, I'm going to take out a -2x y cub / -2x. We're just going to get y cub -2x 4 / that's positive x cub. Now notice what we have here. y cub + x cub. We have that there. And notice that it's equal to 35. So we're going to replace this with have -2x * 35 over y 5th. 2 * 35 is 70. So we're going to get -70x over y 5th. So this corresponds to answer choice D.