# How to find the derivative of y= x^-1/e^-x

• intenzxboi
In summary, the conversation is about finding the derivative of the function y = ((x)^−1) (e^−x). The person attempted to solve it and got [(x^-1) - (x^-2)] e^-x, but the book's answer is −(x^−1 + x^−2)e^−x. They realize that they lost a sign and get the correct answer by differentiating e^-x using the chain rule.
intenzxboi

## Homework Statement

y = ((x)^−1) (e^−x)

## The Attempt at a Solution

Solving this out i got
[(x^-1) - (x^-2)] e^-x

however the book is telling me the answer is
−(x^−1 + x^−2)e^−x

wouldnt that give me (-x^−1 - x^−2)e^−x ??
what did i do wrong??

intenzxboi said:

## Homework Statement

y = ((x)^−1) (e^−x)

## The Attempt at a Solution

Solving this out i got
[(x^-1) - (x^-2)] e^-x

however the book is telling me the answer is
−(x^−1 + x^−2)e^−x

wouldnt that give me (-x^−1 - x^−2)e^−x ??
what did i do wrong??
Apparently you lost a sign somewhere. I'm guessing that you forgot the factor of (-1) when you differentiated e^(-x).
$$d/dx(x^{-1}e^{-x}) = x^{-1} e^{-x} (-1) - x^{-2}e^{-x}$$
$$= -x^{-1}e^{-x} - x^{-2} e^{-x}$$
which agrees with the book's answer.

BTW, you're not "solving this out;" you're differentiating this function or calculating the derivative of the given function.

Thanks i thought the derivative of e^-x stayed the same

intenzxboi said:
Thanks i thought the derivative of e^-x stayed the same

$$\frac{d}{dx}e^x = e^x$$

However, in general (using the chain rule)

$$\frac{d}{dx} e^{f\left(x\right)} = f^\prime\left(x\right)e^{f\left(x\right)}$$

$$f\left(x\right) = -x$$

So

$$\frac{d}{dx}e^{-x} = \frac{d}{dx}\left(-x\right)e^{-x} = - e^{-x}$$

Do you follow?

## 1. What is the basic concept of finding the derivative of y= x^-1/e^-x?

The derivative of a function is the rate of change of that function at a specific point. In other words, it measures how quickly the function is changing at a particular point on the graph. To find the derivative of y= x^-1/e^-x, we use the power rule and the chain rule.

## 2. How do you apply the power rule to find the derivative of y= x^-1/e^-x?

The power rule states that for a function of the form y= x^n, the derivative is given by dy/dx = nx^(n-1). For y= x^-1/e^-x, we can rewrite it as y= (1/x)*(1/e^-x). Using the power rule, we get dy/dx = (-1/x^2)*(1/e^-x). However, we also need to apply the chain rule since the function is nested within another function. This leads us to the final derivative of dy/dx = (-1/x^2)*(1/e^-x)*(-e^-x) = e^-2x/x^2.

## 3. What is the role of the chain rule in finding the derivative of y= x^-1/e^-x?

The chain rule is used when we have a function within another function. In the case of y= x^-1/e^-x, the function e^-x is nested within the function 1/x. The chain rule allows us to find the derivative of the inner function and then multiply it by the derivative of the outer function. In this case, the derivative of e^-x is -e^-x and the derivative of 1/x is -1/x^2. Multiplying these two derivatives gives us -e^-2x/x^2, which is the final derivative of y= x^-1/e^-x.

## 4. Can you explain the steps to find the derivative of y= x^-1/e^-x in simpler terms?

To find the derivative of y= x^-1/e^-x, we first use the power rule to find the derivative of the function x^-1, which is -1/x^2. Then, we use the chain rule to find the derivative of the function e^-x, which is -e^-x. Finally, we multiply these two derivatives to get the final derivative of -e^-2x/x^2.

## 5. Are there any special rules or tricks to remember when finding the derivative of y= x^-1/e^-x?

The key rule to remember is the chain rule, which is essential when dealing with nested functions. Also, it is helpful to remember the power rule, which states that the derivative of x^n is nx^(n-1). It may also be useful to simplify the function before finding the derivative to make the process easier. In this case, we can simplify y= x^-1/e^-x to y= e^x/x.

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