In Exercises 1–24, find the derivative of y with respect to th...

Question

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.

3. y = (1/4)xe^(4x) - (1/16)e^(4x)

Accepted Answer

Welcome back, everyone. Find the derivative of Y with respect to X4 Y equals 13 xe the power of 3X minus 1 divided by 9e to the power of 3X. For this problem, we can begin by factoring out our function. Specifically, let's notice that we can take out a 19th. It is the lower fraction. We can also take out E to the power of 3 X. Now, what remains for the first term, that would be 3 because 3 divided by 9 gives us 1/3, and we also are going to have X. And for the second part, well, we are just racing 1. So the factors form in this case is going to be 1/9 to the power of 3 X multiplied by 3X minus 1. And now we can differentiate Y using the product rule. We are also going to factor out the constant, which is 19th. And now let's understand that we have two functions. It's the power of 3 X is multiplied by 3X minus 1. So we're going to apply the product rule as follows. Let's remember that we are first of all, taking the derivative of the first function, so that's the derivative of it to the power of 3 X. We are multiplying by the second function, which is 3x minus 1. And we are adding the first function eats the power of 3 X multiplied by the derivative of the second one, so the derivative of 3 x minus 1. And now we want to calculate the derivative. So we get 1/9 in parent, the derivative of E to the power of 3 X can be found using the chain rule. We first will get eat to the power of 3 X. And then multiply by the derivative of 3 X, which is 3. We multiply it by 3 X minus 1. And then we add eats the power of 3 X multiplied by 3 x minus 1 derivative, which is equal to 3, right? Because the derivative of -1 is 0, it's a constant, and the derivative of 3 X is 3 because the derivative of X is 1. Now what we can do is simply factor out. 3, so we're going to get 3 divided by 9, which gives us 1/3, and in C, we're going to get E to the power of 3 X. Multiplied by 3 x minus 1. Plus E to the power of 3 X, which essentially shows that we can also factor out E to the power of 3 X. So let's write it in the numerator. And now we are going to end up with 3 X minus 1 + 1, right? And Cs. We can also remove parcs because we don't have any factor in front. So basically have each to the power of 3 X divided by 3, multiplied by 3X minus 1 + 1. We can cancel out -1 and 1. And now 3 divided by 3 gives us 1, right? So we essentially get our final answer. X multiplied by E to the power of 3 X. Let's label our final answer and thank you for watching.

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.3. y = (1/4)xe^(4x) - (1/16)e^(4x)

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In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
3. y = (1/4)xe^(4x) - (1/16)e^(4x)