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Determine the derivative.$f(x)=e^{a x}(x-1 / a), a$ is a constant

$a x e^{a x}$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 4

The Derivative of the Exponential Function

McMaster University

Baylor University

Idaho State University

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alright. So it's important to recognize that A is a constant in this problem because the derivative of any constant, whether it's a or one over a, um, the derivative of either those will give you zero. Okay, that's not a good notation. But that's important to recognize as we're taking the derivative of e to the A X Times X minus one over A. Because if I were doing this problem, it's just the main thing is I would just start with the derivative right away because what you have is a giant product, which means we're gonna do the product. Cool. But then, on the left side, you also have a changeable because we'll just jump into the derivative when you take the derivative of the left side. The derivative of E to the A X is either the a x times, the derivative of a X, which is a and then you leave X alone. Uh, you leave the X minus one over a alone, and then plus, now you leave E to the a X alone times the derivative of the right side to be one. And this is really where the magic happens because as we distribute a E to the X into the problem. I just like to write the X in front of E to the A X. Well, a times one over Able cancels, you're left with one. Eat the a X and then later you have a plus e to the X. That's why it seems like magic. Because these two pieces canceled and you're correct. Answer is just a X e to the a X power. That's it. You could work through this problem a few other ways that this is the, you know, a good method. Um, if you wanted to, you could distribute first. But your stuff to do the product will, no matter how you do the problem. So this is how I'm gonna leave it.

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