Find y'' if:

a. y = csc x

Question

Find y'' if:

a. y = csc x

Accepted Answer

Welcome back, everyone. In this problem, we want to calculate the second derivative of the function F X equals set X. A says it's set X cubed minus X. B says it's set X minus 2 cube X. C 2 cube X minus X, and D 2 cube X plus X. Now, to find the second derivative, essentially what that means is that we're just going to differentiate F of X twice, OK? So let's find the first derivative and then the second in order to solve. So solving for the first derivative. Which we can write as F of X. That means we're trying to find the derivative with respect to X or the 2 of X, which is equal to seXX. Thus, this is the first derivative. Now, we need to differentiate this again to get the 2nd derivative, OK? So no solving for the second derivative of X. OK. That means we're finding the derivative with respect to X of the first derivative. But now the question is how do we do that? Well, here, notice that our first derivative is a product, so we can differentiate by using the product rule. Recall that in the product rule it says that if we have a function Y equals UV where U and V are functions of X, then the derivative of Y equals UV plus U V. In other words, we differentiate both of our functions and then plug them into the product tool. So in this case, OK, if we let. This U, sorry, if we let you be equal to the C of X. And we let v be equal to the tangent of X. Then once we differentiate these, we can plug them in using the product through to solve. No, like we said earlier, the derivative of the C can of X is sec X1 X. OK. And the derivative of a tangent of X is equal to squared X, OK? So now, that means by the product rule, the second derivative is going to be equal to UV, so that's 6 X multiplied by 62 X. Plus UV, so that's gonna be 0 X multiplied by Sec X1 X. Now, we can do some simplifying here because 6 X multiplied by 6 squared X equals secubed X, OK. And 1 X multiplied by 10 X is 100 square X, OK? So that's gonna be 6 X multiplied by 10 squared X. We can take it a step further by factoring out to C count of X, which is gonna give us 6 X multiplied by 62 X plus 102 X. Thus, this is the 2nd derivative of the C of X. If we look back at our answer choices, OK, well, you know what? We could even take it a step further. We we're not finished yet. We know that the 62 X plus 102 X is an identity. We know that it equals +262 x minus 1, OK? So then that means our second derivative is gonna be equal to set X multiplied by +262 x minus 1, which is gonna be equal to 2 cubed X minus the C kind of X. And this is our final answer, yeah. So now, when we look back at our answer twice, then this tells us that C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Find y'' if:

a. y = csc x

Key Concepts

Second Derivative

Cosecant Function

Differentiation Rules

Watch next

Find y'' if:a. y = csc x

Key Concepts

Second Derivative

Cosecant Function

Differentiation Rules

Watch next

Find y'' if:

a. y = csc x