[Solved] What is the Derivative of e^1/x?

The derivative of e^1/x is $-\frac{1}{x^2} e^{1/x}$. So if y=e^1/x, then dy/dx = -1/x² 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/x² 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/x² 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.

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