The derivative of square root of e^{x} is $\frac{1}{2}\sqrt{e^x}$. Here, we will find the derivative of root e to the power x by the chain rule of derivatives.

## What is the Derivative of root e^{x}?

We know that square root of x can be written as x^{1/2}. So we have

√e^{x} = (e^{x})^{1/2} = e^{x/2}.

∴ $\dfrac{d}{dx}$(√e^{x}) = $\dfrac{d}{dx}$(e^{x/2})

= e^{x/2 }$\dfrac{d}{dx}(\dfrac{x}{2})$

= e^{x/2 }$\times \dfrac{1}{2}$

= $\dfrac{1}{2} \sqrt{e^x}$

So the derivative of root e to the power x is (1/2)√e^{x}.

## Derivative of root e^{x} by Chain Rule

As square root of e^{x} is a function of functions, we need to find the derivative of √e^{x} using the chain rule of derivatives.

Let us put z=e^{x}.

Differentiating with respect to x, we obtain that $\dfrac{dz}{dx}=e^x$. Now, using the chain rule, the derivative of √e^{x}, that is, d/dx(√e^{x}) is equal to

$\dfrac{d}{dx}(\sqrt{e^x})$

= $\dfrac{d}{dz}(\sqrt{z}) \times \dfrac{dz}{dx}$

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

= $\dfrac{1}{2} z^{\frac{1}{2}-1} \times e^x$ by the prower rule of derivatives: $\dfrac{d}{dx}(x^n)$ = nx^{n-1}.

= $\dfrac{1}{2} z^{-\frac{1}{2}} e^x$

= $\dfrac{1}{2} e^{-\frac{x}{2}} e^x$ as z=e^{x}.

= $\dfrac{1}{2} e^{x-\frac{x}{2}}$

= $\dfrac{1}{2} e^{\frac{x}{2}}$

= $\dfrac{1}{2} \sqrt{e^x}$ as we know that x^{1/2}=√x.

So the derivative of square root of e^{x} is equal to (1/2)√e^{x} and this is obtained by the chain rule of derivatives.

**Also Read**

**Derivative of root sinx by First Principle**

**Derivative of e ^{3x} by First Principle**

## FAQs

**Q1: What is the Derivative of root e ^{x}?**

Answer: The derivative of the square root of e^{x }is equal to 1/2 √e^{x}.