Derivative exponential problem

In summary, the derivative of the function (1-2x)e^{-x} can be found using either the product rule or the quotient rule. However, in using the product rule, one must remember to use the chain rule in the second term, resulting in a different answer than when using the quotient rule. The correct answer is -e^{-x} - 2x e^{-x} or e^{-x}(-1-2x).
  • #1
MacLaddy
Gold Member
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Homework Statement



Compute the derivative of the following function

[itex](1-2x)e^{-x}[/itex]

Homework Equations



Product Rule and Quotient rule

The Attempt at a Solution



My problem here is that I come up with two different answers when I use the quotient rule vs. the product rule.

Trying it with the product rule

f(x)= (1-2x)e^-x
f'(x)= -2e^-x + e^-x(1-2x)
f'(x)= e^-x(-2+(1-2x))
[itex]f'(x)= e^{-x}(-1-2x)[/itex] or [itex]\frac{-1-2x}{e^x}[/itex]

With the quotient rule

f'(x)= (e^-x(1-2x) - (-2e^-x)) / [e^-x]^2
f'(x)= (e^-x-2e^-x+2e^-x) / [e^-x]^2
f'(x)= (e^-x(-2x+2)) / [e^-x]^2
[itex]f'(x)= \frac{-2x+2}{e^{-x}}[/itex] or [itex]e^x(-2x+2))[/itex]

As you can see, these are two different answers. I would think that I should have the same solution either way I do this, so what am I doing wrong?
 
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  • #2
MacLaddy said:

Homework Statement



Compute the derivative of the following function

(1-2x)e^(-x)

Homework Equations



Product Rule and Quotient rule

The Attempt at a Solution



My problem here is that I come up with two different answers when I use the quotient rule vs. the product rule. ( And that I can't figure out the Latex for e^-x... If someone can show me how to do that I'll clean this post up)
[itex]e^-x[/itex]

Trying it with the product rule

f(x)= (1-2x)e^-x
f'(x)= -2e^-x + e^-x(1-2x)
f'(x)= e^-x(-2+(1-2x))
f'(x)= e^-x(-1-2x) or (-1-2x)/e^x

With the quotient rule

f'(x)= (e^-x(1-2x) - (-2e^-x)) / [e^-x]^2
f'(x)= (e^-x-2e^-x+2e^-x) / [e^-x]^2
f'(x)= (e^-x(-2x+2)) / [e^-x]^2
f'(x)= (-2x+2) / (e^-x) or e^x(-2x+2)

As you can see, these are two different answers. I would think that I should have the same solution either way I do this, so what am I doing wrong?

I don't get either of your answers when I take the derivative.

(1-2x)e^{-x} = e^{-x} -2xe^{-x} = -e^{-x} -2(product rule) = ??
 
  • #3
MacLaddy said:
... ( And that I can't figure out the Latex for e^-x... If someone can show me how to do that I'll clean this post up)

[itex]e^-x[/itex]

...
Put the whole exponent in braces { } .

[itex ]e^{-x}[/itex ] gives   [itex]e^{-x}[/itex]
 
  • #4
Thanks SammyS. I cleaned up some of the above. Hopefully that helps.
 
  • #5
MacLaddy said:
Trying it with the product rule

f(x)= (1-2x)e^-x
f'(x)= -2e^-x + e^-x(1-2x)
Here's the problem. It should be a minus above, not a plus. (Remember, chain rule.)
MacLaddy said:
With the quotient rule

f'(x)= (e^-x(1-2x) - (-2e^-x)) / [e^-x]^2
Whoa. If you want to use the quotient rule, then you need to rewrite f(x) as
[tex]f(x)=\frac{1-2x}{e^x}[/tex]
... so there shouldn't be any e-x in the first step:
[itex]f'(x) = \frac{e^x(1-2x) - (-2e^x)}{(e^x)^2}[/itex]
= ...
 
  • #6
Thanks eumyang. I'm actually seeing numerous errors that I've typed above, and I'm working on cleaning it up with Latex.

Please stand by until I can clean up this mess. :smile:
 
  • #7
Ahh, I didn't need to completely retype that. You are perfectly correct, eumyang, I was making those mistakes on both. I wasn't considering that the derivative of [itex]e^{-x}[/itex] is [itex]-e^{-x}[/itex]. And I wasn't playing with the signs right on the quotient rule.

Thank you very much, I appreciate the help.
 

1. What is a derivative exponential problem?

A derivative exponential problem refers to a mathematical problem that involves finding the derivative of an exponential function. This means finding the rate of change of the function at a specific point.

2. How do you solve a derivative exponential problem?

To solve a derivative exponential problem, you can use the basic rules of differentiation, such as the power rule and the chain rule. It is also helpful to have a thorough understanding of exponential functions and their properties.

3. What is the purpose of solving a derivative exponential problem?

Finding the derivative of an exponential function can help us understand the behavior of the function, such as its rate of growth or decay. It is also a useful tool in many fields of science and engineering, as it allows us to make predictions and analyze data.

4. Can you give an example of a derivative exponential problem?

One example of a derivative exponential problem is finding the derivative of the function f(x) = e^x. The solution to this problem is f'(x) = e^x, which represents the instantaneous rate of change of the function at any given point.

5. Are there any real-world applications of derivative exponential problems?

Yes, there are many real-world applications of derivative exponential problems. For instance, in finance, the compound interest formula can be modeled using an exponential function, and finding the derivative of this function can help determine the rate of growth of investments. In physics, exponential functions are used to model radioactive decay, and finding the derivative can help predict the rate of decay.

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