51–56. Second derivatives Find d²y/dx².
sin x + x²y...

Question

51–56. Second derivatives Find d²y/dx².

sin x + x²y =10

Accepted Answer

Welcome back, everyone. Calculate the second derivative of cosine of X plus XY cubed equals 59. For this problem, let's apply the concept of implicit differentiation. So we're going to take the left-hand side. Cosine of X is added to X cubed Y. We're going to take the derivative of that and we're going to set it equal to the derivative of the right hand side, so the derivative of 59. Now on the left hand side, we first will differentiate cosine of X. Let's recall that the derivative of cosine is negative sign, so we're getting negative sign of X, and now we want to differentiate the product. Specifically, we want to differentiate X cubed Y. Based on the product rule, we take the derivative of X cubed, multiplied by Y and add x cubed multiplied by y derivative, which gives us 3 x 2y plus x cubed Y. So here we are applying the power rule, and we can write those terms in our equations. We're going to get 3 X2 Y plus x cubedy. And this is equal to 0 because the derivative of a constant is 0. So we have our first equation and we can solve for the first derivative if we want, or we can immediately perform another step of implicit differentiation. So let's go ahead and differentiate the result and the expression once again. The derivative of sinus cosine. So we're going to get the derivative of negatives of X, which is equal to negative cosine of X. Now, once again, we want to differentiate a product between X2 and Y because we can factor out the 3. So what is the derivative of X2 Y? Well, this is equal to the derivative of X2. Multiplied by y plus x2y derivative based on the product rule which is 2 x y plus x2y and now from here we will be able to substitutey from the original equation right? so let's Keep that in mind, but for now we can just say that this is. We multiplied by 2 X Y. Plus x squared y. And then we are adding. The derivative of X cubedy. Based on the product rule, we're going to take the derivative of X cubed multiplied by Y plus x cubed multiplied by the derivative of Y, which is the second derivative. So let's write down this term. Specifically, that's 3 X where the Y plus X cubed Y. So once gains 3 X 2 Y plus X cubed Y and this is equal to 0. Our goal is to solve for the second derivative, right? So we want to isolate the final term. Let's move all of the other terms to the right. We're going to swap those signs. So we get cosine of X plus 6 XY. Now is it plus, well, essentially we have to swap that positive sign into a negative sign, right? Because we are. Changing the position of the second term, so minus 6. X Y Then we are subtracting 3x2y. we are performing distribution and then we are subtracting 3 x 2y. We also have to understand that the remaining term still has X cubed in front, right. So if we want to solve for Y prime, we have to divide both sides by X cubed. And we can see that essentially because if we consider the domain of the first derivative, we are going to see that X cannot be equal to 0, right? So, essentially, We are allowed to divide by X since we have a restriction from the first derivative. We will soon see why this is the case. For now, let's not worry about the domain. We have the general expression of the second derivative, and what we can see is that we can combine the terms -3 x2y and another 3 x 2. So Y becomes cosine of x minus 6 X Y minus 6 X 2 Y. All of that is divided by x cubed. Our goal is to solve for y so we can see the original equation. And now we are going to move sine x to the right. This gives us sin x. We're going to move V X squad to the right, so we're changing the sign from positive to negative. So sin x minus 3 X 2 Y. And so, so for we need to divide both sides by X cube. Well done. And now we're going to substitute this into the the expression of the second derivative. So we get cosine of X minus 6 XY. Minus 6 x squared multiplied by. Sign of x minus 3 x 2y divided by X cubed. So here we have our first derivative notice that the domain is X not equal to 0, right? And we are dividing. The whole expression. B X cubed. Now, X2 divided by X cubed, well, this gives us 1 divided by X, so we can cross out X2, and we're going to get X in the denominator. And now from here, well, we can find the common denominator because the first two terms. Don't have any denominator, so let's multiply both of them by x. This gives us y equals x cosine x minus 6 X squared y. We're multiplying by x once again. We are then subtracting sin x. Which is multiplied by 6, so 6 sin x, and then we are adding. -6 multiplied by -3. This gives us 18. X squared y Now our denominator is X, and then there's another denominator which is X cubed. So this double division gives us an overall exponent of 4, right? We have X, the power of force, and that's nearly it. We just want to combine the like terms. So we can continue and this gives us X cosine of X. Minus 6 + 18 gives us 12. X squared y Let's make sure that we can properly see the expressions. So we get X cosine of X plus 12 x squared y, and the final term is minus. 6 X. And all of that is divided by x to the power of 4th. And this is our 2nd derivative. Thank you for watching.

51–56. Second derivatives Find d²y/dx².
sin x + x²y =10

Key Concepts

Implicit Differentiation

First Derivative

Second Derivative

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