Second derivatives Find d²y/dx²for the following functions.

Question

y = x cos x²

Accepted Answer

Welcome back, everyone. Determine the second derivative for the function Y equals x2 tangent of X cubed. Now to identify the second derivative, we first of all want to identify the first derivative. So here we have a product of two functions. The first one is X2d. Let's call it G, and then we have F which is tangent of X cubed. So now what we're going to do is simply differentiate each. Now G is going to be 2 X based on the power rule. And now for the F, well, we want to evaluate the derivative of tangent, which is c 2d. This is from the table of basic derivatives. We keep the angle the same that's x cubed, and then based on the chain rule, we have to multiply by the derivative of X cubed, which essentially gives us 3 x 2 squad of x cubed. And now what we have to do is simply recall the product rule. So essentially Y is going to be equal to G multiplied by F plus G multiplied by F. So in this case we're going to take G, which is to X, multiply that by F which is tangent. Of X cubed and we're going to add G which is x2 multiplied by F. So we get 3x2 multiplied by x2 which gives us 3 X to the power of 4th multiplied by 2 squad of X cubed, and that's our first derivative. Now, when we want to identify the second derivative. We're going to take the derivative of the first derivative, and this means that now we have to apply the product rule for each term. So we can begin with the first one. We can just write that we factor out the constant, that's 2, and then we take the derivative of the first function. So that's X multiplied by the second function, tangent X cubed. And then we're simply adding x multiplied by the derivative of tangent of X cubed. So we are applying the same product rule and then for the second term we can factor out the 3, and then we're taking the derivative of the first function. Multiplied by the second function which is square x cubed, and we have to add x2 multiplied by the derivative of squad x cubed. So now we have defined the rules, and now we can evaluate the derivatives. So first of all, we have 2, the derivative of X is one we end up with tangent of X cubed. Plus x multiplied by the derivative of tangent with respect to x cubed, right, so that's 2 squared x cubed and now we have to apply the chain rules we are multiplying that by the derivative of X cubed which is 3 x 2d based on the power rule. And now for the 2nd term we have 3. Now the derivative of X to the 4th is 4 x cubed based on the power rule. This is multiplied by squad of x cubed, and then we are adding x4 multiplied by now the derivative of squad based on the power rule because we have squad right, will be 22 x cubed. And then we have to apply the chain rule for the inner function. The derivative of s and is s and tangent, so we get 2 X cubed, tangent X cubed, and then finally we have to multiply by the derivative of X cube, which is 3 x 2. What we have to do from here is essentially distribute the terms. So first of all, we get 2 tangent X cubed plus 2 multiplied by 3, which gives us 6. X cued because we have X multiplied by X2, and then we have squared X cubed. And now we're looking at the second set of recs we get plus 12 x cubed 2 squared x cubed, and then if we look at the final term, well, we have 2 multiplied by 3, so that's plus 6. Then we have X to the power of 4th, and then we have sequent multiplied by 2 q, which gives the sequent to the power of 4th of Actually, we have sequence of X cube, right? So let's be careful here. So that'd be 3 multiplied by 2 gives us 6. Then we have X2d, right? So we have to modify that. And X squared multiplied by x the power of 4th gives us X to the power of 6. We can write x the power of 63 multiplied by 2 multiplied by 3 gives us 18. And then considering the sequent function, we have sequent of X cubed multiplied by sequent of X cubed. So that'd be sequent squared of X cubed, right? And then we have tangent. Of X cubed. And now we are going to combine the like terms. So we got two tangent of X cubed. Now let's consider the second set. We have 6 X cubed squared X cubed plus 12 x cubed square x cubed. So these can be added into 18 x cubed square x cubed, and then finally we have another term which is 18 X the power of sixth. A 2nd squared of x cubed tangent of X cubed and that will be our final answer. So let's label it and thank you for watching.

Second derivatives Find d²y/dx²for the following functions.
y = x cos x²

Key Concepts

Second Derivative

Product Rule

Chain Rule

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