The derivative of e^{1/x} is $-\frac{1}{x^2} e^{1/x}$. So if y=e^{1/x}, then dy/dx = -1/x^{2} e^{1/x}. In this post, we will learn how to differentiate e to the power 1/x.

## Derivative of e^{1/x}

**Question:** What is the derivative of e raised to 1/x?

*Answer:* The derivative of e raised to 1/x is equal to $-1/x^2 e^{1/x}$.

**Explanation:**

We will find the derivative of $e^{\dfrac{1}{x}}$ by the chain rule method of derivatives. The following steps to be followed.

**Step 1:** Let us put z=1/x

**Step 2:** We have $\dfrac{dz}{dx}=-1/x^2$

**Step 3:** Then by the chain rule, the derivative of e to the power 1/x^2 is given by

$\dfrac{d}{dx}(e^{1/x^2})$ $=\dfrac{d}{dz}(e^z) \times \dfrac{dz}{dx}$

$=e^z \times (-\dfrac{1}{x^2})$

$=-\dfrac{1}{x^2} e^{1/x}$ as we have z=1/x.

**Conclusion:** So the derivative of e^{1/x} is equal to $-\dfrac{1}{x^2}$ e^{1/x}, that is, d/dx(e^{1/x}) = -1/x^{2} e^{1/x}.

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## Solved Problems

**Question:** Find the derivative of e^{1/x} at x=1.

**Answer:**

From above, the derivative of e^{1/x} is given as follows: d/dx (e^{1/x}) = -1/x^{2} e^{1/x}. At the point x=1, this derivative will be equal to

$\Big[\dfrac{d}{dx}(e^{1/x}) \Big]_{x=1}$

= $\Big[ -\dfrac{1}{x^2} e^{1/x} \Big]_{x=1}$

= $-\dfrac{1}{1^2} e^{1/1}$

= -e.

So derivative of e^{1/x} at x=1 is equal to -e.

## FAQs

**Q1: What is the derivative of e ^{1/x}?**

**Answer:** The derivative of e^{1/x} is equal to $-\frac{1}{x^2}$ e^{1/x}.