What is the derivative of y = e^kx?

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Accepted Answer

Hi, everyone, let's take a look at this practice problem dealing with derivatives. This problem says to find the derivative of the function H of Z, which is equal to E raised to the 5 Z. We're given 4 possible choices as our answers. For choice A, we have 5 multiplied by E raised to the Z. For choice B, we have E raised to the 5 Z. For choice C, we have E raised to the 5 Z divided by 5. And for choice D, we have 5 multiplied by E raised to the 5 Z. Now, in order to take the derivative of this function, we're going to want to use the chain rule. And so we recall for the chain rule that we have the derivative of Y. With respect to Z in this case, it's going to be equal to the derivative of Y with respect to U, multiplied by the derivative of U with respect to Z. So, if we look at our expression for H of Z, we're going to want to set R U. Equal to 5 Z. And so we can rewrite our function H in terms of view, so we'll have each of you. This equal toe raises to the U. And so, I'm going to want to take a derivative of you and each of you. And so we're gonna take a derivative. Of you with respect to Z. And this is going to be equal to the derivative with respect to Z of the quantity of 5 Z. And when I take that derivative of 5 Z, this is just gonna be equal to 5. Now, we're gonna want to take our derivative of H with respect to you. And this is gonna be equal to the derivative with respect to you. Of the quantity of E rays to the U. Which is just equal toe raises to the U. So now that I have both of these derivatives calculated, I can use our chain rule to find our derivative of our function H of Z with respect to Z. So our derivative of H with respect to Z. It's going to be equal to or derivative of H which is with respect to you, which is ease to the u. So we'll have E, and here for you we'll substitute back in 5 Z. So we'll have E raised to the 5 Z. And this could be multiplied by. Are derivative of u with respect to z which we calculated as 5. And so we can just simplify this expression, and this is going to be equal to 5, multiplied by E raised to the 5 Z. So this is the answer that I'm looking for. And if I look at my answer choices, this corresponds to answer choice D. So I hope that this explanation has been useful, and I'll see you in the next video.

What is the derivative of y = e^kx?

Key Concepts

Derivative

Exponential Functions

Constant Coefficient in Derivatives

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