Question

Good Question ?
____________

Differentiate
$e {}^{ \sqrt{3x} }$
with respect to x. ​

Answer 1

Answer:

This differentiation is done by the method of Chain Differentiation.

First we differentiate the exponential (e) function. Later we differentiate the power with root alone. Finally we differentiate the term inside the root.

To obtain the final derivative, all the three answers (derivatives of each function) are multiplied.

Derivatives of some important functions:

$\dfrac{d}{dx} (e^x) = e^x\\\\\\\dfrac{d}{dx} (\sqrt{x}) = \dfrac{1}{2\sqrt{x}}\\\\\\\dfrac{d}{dx} (cx) = c \:\:\: \text{('c' is a constant)}$

Differentiating the given question we get:

$\dfrac{d}{dx} (e^{\sqrt{3x}}) = e^{\sqrt{3x}} \times \dfrac{1}{2\sqrt{3x}} \times 3\\\\\\\boxed{ \bf{ \dfrac{d}{dx} (e^{\sqrt{3x}}) = \dfrac{3e^{\sqrt{3x}}}{2\sqrt{3x}}}}$

Answer 2

$\Large{\underbrace{\underline{\bf{ANSWER\:}}}}:$

${\bf{Given\::}}$

• $\bf{e^{ \sqrt{3x}} }$

$\\ {\bf{To\: Find\::}}$

• Differentiation $\bf{e^{ \sqrt{3x}} }$ with respect to x.

$\\ {\bf{Solution\::}}$

Let,

• $\tt{y\:=\:e^{ \sqrt{3x}} }$

Now,

Differentiation the above equation w.r.t x, we get

$:\implies\:\tt{\dfrac{d}{dx}(y)\:=\:\dfrac{d}{dx}\left(e^{ \sqrt{3x}}\right)\:}$

$:\implies\:\tt{\dfrac{dy}{dx}\:=\:\dfrac{d\left(e^{ \sqrt{3x}}\right)}{d\left(\sqrt{3x}\right)}\times{\dfrac{d\left(\sqrt{3x}\right)}{d\left({3x}\right)}}\times{\dfrac{d(3x)}{dx}}\:}$

$:\implies\:\tt{\dfrac{dy}{dx}\:=\:e^{ \sqrt{3x}}\times{\dfrac{1}{2\sqrt{3x}}}\times{3}\:}$

$:\implies\:\bf\pink{\dfrac{dy}{dx}\:=\:\dfrac{3\:e^{ \sqrt{3x}}}{2\sqrt{3x}}\:}$

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Some Important Properties :

⠀⠀ ❶ $\bf{\dfrac{d}{dx}\left(e^{\sqrt{x}}\right)\:=\:e^{\sqrt{x}}\:}$

⠀⠀ ❷ $\bf{\dfrac{d}{dx}\left(\sqrt{x}\right)\:=\:\dfrac{1}{2\:\sqrt{x}}\:}$

⠀⠀ ❸ $\bf{\dfrac{d}{dx}\left({x}\right)\:=\:1\:}$