Find y'' for the following functions.
y = e^x ...

Question

Find y'' for the following functions.

y = e^x sin x

Accepted Answer

Welcome back, everyone. In this problem, we want to find a second derivative Y for the function Y E equals 11 E X plus X. A says it's negative 22 E X X X, B, 22 E X X, C 11 E X X X, and D11E 2 X X. Now if we're gonna find the second derivative, Y, it essentially means that we're going to differentiate Y twice. So let's find the first derivative and differentiate that to get the second derivative. Now notice that Y is a product. 11 E X plus X. What do we know about differentiating products? Well, recall. That if we have a function Y that's equal to UV where U and V are differentiable. Then the derivative of Y is equal to UV plus UV. OK. In other words, we can differentiate U and V and then plug them into the product rule of differentiation to get the derivative of Y. No, we could let any of these terms be equal to U and V. So in this case, let's let you, let me put this in red here for the first derivative. Let's let you be equal to 11 eat the X. And Z equal the cosine of X. Now, the derivative of 11 e to the X is going to be 11E X because remember we differentiate the power and bring it down and the derivative of X is 1. Multiply that by the original. That's how we end up back with 11 E to the X. On the other hand, for V equals the cosine of X, the derivative of X is negative sin X. Therefore, since we have our derivatives, we can apply the product rule, which means it's going to be UV. 11 e to the X multiplied by the cosine of X. Plus UV, the 11 e to the X multiplied by negative sin X or. -11 e to the X X. Notice both terms have 11 e to the X in common, so if we factor it out, we're left with 11 e to the X multiplied by the cosine of X minus sine X. This is the first derivative of Y. Now that we are the first derivative, we can differentiate it again to find the second derivative, OK? So, solving for Y, the second derivative, OK. Which is equal to the derivative of the derivative of Y, the first derivative. Then notice here when we look at the derivative, or sorry, when you look at the first derivative, again, we have two terms multiplying each other. So now we can apply the product rule again, OK? So by applying the product tool. By applying the product rule. Where in this case, we can let you be E 11 E to the X and V becu X minus sine X, OK. Then. If you is 11 to the X, the derivative of you is also 11E to the X. And if V is the cores sign of X minus sine X, then the derivative of V. Is going to be equal to negative sin x minus the cosine of X, because the derivative of sin X is class X. Now that we have that, then by the product rule, that means the second derivative will be equal to UV, so that's 11 E X. Multiplied by X minus sine x, OK. Plus 11 E to the X. Multiplied by a negative sin x minus X. Let's go ahead and try to simplify our problem, OK? Now notice both of them have a factor of 11 e to the X. So if we factor it out. Then we're going to be left with. The cosine of X minus the sign of X minus the sign of X minus the cosine of X. Cosine of X minus the cosine of X equals 0. So we have 11 E X minus sine x minus sin X, which is -2 sin X. No, 11 multiplied by -2 is -22. So the second derivative is gonna be -22 EX multiplied by sin X. When we look back on our answer choices, that tells us that A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Find y'' for the following functions.
y = e^x sin x

Key Concepts

Higher-Order Derivatives

Product Rule

Chain Rule

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