In Exercises 7–26, find the derivative of y with respect to x,...

Question

In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.

y = e^(-5x)

Accepted Answer

Welcome back, everyone. Compute the derivative of Y with respect to X4 Y equals e to the power of x divided by 4. A 1/4e the power of X divided by 4, B4E raises to the power of X divided by 4, C E to the power of X divided by 4, and D 4E to the power of 4 X. For this problem, let's recognize that Y is a composite function, right? Essentially, it is a function e to the power of X, where X is replaced with X divided by 4. So we have an outer function and an inner function. So if we want to identify Dy divided by DX and if we set U equals x divided by 4, which is our inner function, We are going to use the chain rule. Remember that first of all, we are differentiating our function with respect to you. And multiplying by the derivative of U with respect to X. We can begin by differentiating. Our function, which is now Y equals E, raises the power of U where U is equal to X divided by 4, right? So D Y divided by D U. Is going to be the derivative of e to the power of you, which is e to the power of you. You know, because C is equal to x divided by 4, we can already express it in terms of X, that's e to the power of X divided by 4. And now the derivative of U with respect to X is the derivative of X divided by 4. We can factor out 1/4, which is a constant, and multiply by the derivative of X, which is 1, so we simply get 1/4. And now substituting into the chain rule, we're going to get 1/4 multiplied by E to the power of X divided by 4. This corresponds to the answer choice A. Thank you for watching.

In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.y = e^(-5x)

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In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = e^(-5x)