Find f′(x) if f(x) = 15e^3x.

Question

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with derivatives. This problem says determine F of X or the function F of X is equal to 10 multiplied by E raised to the 5 X. We're given 4 possible choices as our answers. For choice A, we have 10 multiplied by E raised to the 5 X. For choice B, we have 50 multiplied by E raised to the 5 X. Choice C, we have 10 multiplied by E to the X. And for choice D, we have 50 multiplied by E to the X. Now, in order to find this derivative of uh which is what we're asked to find in this problem, we will actually need to use the chain rule. So recall for the chain rule that will have the derivative of Y, with respect to X. And that's going to be equal to the derivative of Y with respect to you. Multiplied by the derivative of U with respect to X. So, for our function here, we're gonna set U equal to 5 X. And then that means that We can write our function in terms of you, so we'll have F of you. is equal to 10, multiplied by E to the U. So now we're gonna need to take derivatives of both of these functions. So, when I take the derivative of you with respect to X, That's going to be equal to the derivative with respect to X. Of 5 X. Which is equal to 5. For our other function f of you will need to take its derivative. With respect to you, And this is going to be equal to. The derivative with respect to you. Of the quantity of 10 multiplied by E rates to the U. Which is equal to 10 multiplied by E raised to the U. So recall that the derivative. With respect to you, Of E U is equal to eu. So now that we have these two derivatives calculated, we can actually use a chain rule to find RF. So, F. Of X, which is the derivative of X or derivative of F with respect to X, and that's going to be equal to our derivative of F with respect to U, which we just calculated as 10 multiplied by E to the U. So, we'll have 10 multiplied by E, and in place of you we'll go ahead and substitute back in our 5 X, so there'll be E raised to the 5 X. And then that's going to be multiplied by our derivative of U with respect to X, which is equal to 5. And so we can simplify this expression, and this will be equal to 50, multiplied by E raised to the 5 X. So that is the answer that we're looking for. And if we look at our answer choices, correct answer choice corresponds to answer choice B. So I hope that this explanation has been useful, and I'll see you in the next video.

Find f′(x) if f(x) = 15e^3x.

Key Concepts

Derivative

Exponential Functions

Chain Rule

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