# Derivative of Square Root of e^x: Proof by Chain Rule

The derivative of square root of ex 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 ex?

We know that square root of x can be written as x1/2. So we have

√ex = (ex)1/2 = ex/2.

∴ $\dfrac{d}{dx}$(√ex) = $\dfrac{d}{dx}$(ex/2)

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

= ex/2 $\times \dfrac{1}{2}$

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

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

## Derivative of root ex by Chain Rule

As square root of ex is a function of functions, we need to find the derivative of √ex using the chain rule of derivatives.

Let us put z=ex.

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

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

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

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

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

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

= $\dfrac{1}{2} \sqrt{e^x}$ as we know that x1/2=√x.

So the derivative of square root of ex is equal to (1/2)√ex and this is obtained by the chain rule of derivatives.

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## FAQs

Q1: What is the Derivative of root ex?

Answer: The derivative of the square root of ex is equal to 1/2 √ex.