Question

$\displaystyle\maltese \underline{\underline{\textbf{\textsf{\pink{Question}}}}}$
Find the derivative for $\displaystyle\sf \dfrac{e^{^3\!/_2}}{x^2}$

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Answer 1

Solution:-

=> To find

$\to\sf \: Derivative \: \: of \: \: \dfrac{ {e}^{ \frac{3}{2} } }{ {x}^{2} }$

=> We can write as

$\sf \to \: \dfrac{d }{dx} \: \bigg(\dfrac{e {}^{ \frac{3}{2} } }{ {x}^{2} } \bigg)$

$\sf \to \: \dfrac{d \: }{dx} ({e}^{ \frac{3}{2} } \times {x}^{ - 2})$

We know that

$\sf \to \: {e}^{ \frac{3}{2} } \: \: is \: \: constant \:$

We use this property

$\sf \to\: \dfrac{dax}{dx} \: \: \: \: \: where \: a \: is \: constant \: \implies \: a\dfrac{dx}{dx}$

$\sf \to \: {e}^{ \frac{3}{2}} \dfrac{d {x}^{ - 2} }{dx}$

$\sf \to \: {e}^{ \frac{3}{2} } \times - 2x {}^{ - 2 - 1}$

$\sf \to \: {e}^{ \frac{3}{2} } \times - 2x {}^{ - 3}$

We can write as

$\sf \to \: \dfrac{ - 2 {e}^{ \frac{3}{2} } }{ {x}^{3} }$

Answer is

$\sf \to \: \dfrac{ - 2 {e}^{ \frac{3}{2} } }{ {x}^{3} }$

Some property of differentiation

$\sf \to \: \dfrac{d}{dx} (sinx) = cosx$

$\sf \to \: \dfrac{d}{dx} ( - cosx) = sinx$

$\sf \to \: \dfrac{d}{dx} (tanx) = sec {}^{2} x$

$\sf \to \: \dfrac{d}{dx} ( - cotx) = cosec {}^{2} x$

$\sf \to \: \dfrac{d}{dx} (secx) = secxtanx$

$\sf \to \: \dfrac{d}{dx} {e}^{x} = {e}^{x}$

$\sf \to \: \dfrac{d}{dx} (log |x| ) = \dfrac{1}{x}$

Answer 2

$\underline{\underline{\sf{ Derivative\:\:of \:\:}\sf{\displaystyle\frac{e^{\tfrac{3}{2}}}{x^2}}}}$

We can also write it as :

$\to\sf{\displaystyle \frac{d}{dx}\left(\frac{e^{\tfrac{3}{2}}}{x^2}\right)}$

Take the Constant Out : (a × f)' = a × f'

$\to\sf{\displaystyle e^{\tfrac{3}{2}}\frac{d}{dx}\left(\frac{1}{x^2}\right)}$

Apply Exponent Rule : 1/a = a⁻¹

$\to\sf{\displaystyle e^{\tfrac{3}{2}}\frac{d}{dx}\left(x^{-2}\right)}$

Apply Power Rule : $\sf{\displaystyle\frac{d}{dx}\left(x^a\right)=a\times x^{a-1}}$

$\to\sf{\displaystyle e^{\tfrac{3}{2}}\left(-2x^{-2-1}\right)}$

Simplify :

$\to\sf{\displaystyle e^{\tfrac{3}{2}}\left(-2x^{-3}\right)}$

$\to\sf{\displaystyle -2e^{\tfrac{3}{2}}x^{-3}}$

Apply Exponent Rule : a⁻ᵇ = 1/aᵇ

$\to\sf{\displaystyle 2e^{\tfrac{3}{2}}\frac{1}{x^3}}$

$\to\sf{\displaystyle -\frac{1\cdot \:e^{\tfrac{3}{2}}\cdot \:2}{x^3}}$

$\boxed{\boxed{\to\sf{-\dfrac{2e^{\tfrac{3}{2}}}{x^3}}}}$