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

 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.

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