Question

Find derivation of : (x^2+3x^3+ 4x^2+2)/x^3
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Answer 1

Given

$\to\sf \dfrac{x^2+3x^3+4x^2+2}{x^ 3} \: differentiate\: with\:respect\:to\:x$

We can write as

$\to\sf d\dfrac{\bigg(\dfrac{x^2+3x^3+4x^2+2}{x^3}\bigg) }{dx}$

Now Simplify the Equation

$\to\sf \dfrac{x^2+3x^3+4x^2+2}{x^3} =\dfrac{x^2}{x^3} +\dfrac{3x^3}{x^3} +\dfrac{4x^2}{x^3} +\dfrac{2}{x^3}$

$\to\sf\dfrac{1}{x} +\dfrac{3}{1} +\dfrac{4}{x} +\dfrac{2}{x^3}$

$\to\sf x^{-1}+3+4x^{-1}+2x^{-3}$

$\to\sf \dfrac{d(\sf x^{-1}+3+4x^{-1}+2x^{-3})}{dx}$

$\to\sf \dfrac{d(x^{-1})}{dx} +\dfrac{d(3)}{dx} +\dfrac{d(4x^{-1})}{dx} +\dfrac{d(2x^{-3})}{dx}$

$\to\sf\dfrac{d(x^{-1})}{dx} +\dfrac{d(3)}{dx} +4\dfrac{d(x^{-1})}{dx} +2\dfrac{d(x^{-3})}{dx}$

$\to\sf -1x^{-1-1}+0+4\times-1x^{-1-1}+2\times-3x^{-3-1}$

$\to\sf -1x^{-2}-4x^{-2} -6x^{-4}$

$\to\sf\dfrac{-1}{x^2} -\dfrac{4}{x^2} -\dfrac{6}{x^4}$

Answer

$\to\sf\dfrac{-1}{x^2} -\dfrac{4}{x^2} -\dfrac{6}{x^4}$

Answer 2

Answer:

$\frac{ {x}^{2} + 3 {x}^{3} + 4 {x}^{2} + 2 }{ {x}^{3} } \\ \\ we \: can \: write \: as \\ : \implies \: d \frac{ \frac{ {x}^{2} + 3 {x}^{3} + 4 {x}^{2} + 2 }{ {x}^{3} } }{dx} \\ \\ simplify \: \: the \: \: eqation \\ : \implies \: \frac{ {x}^{2} + 3 {x}^{3} + 4 {x}^{2} + 2 }{ {x}^{3} } = \frac{ {x}^{2} }{ {x}^{3} } + \frac{3 {x}^{3} }{ {x}^{3} } + \frac{4 {x}^{2} }{ {x}^{3} } + \frac{2}{ {x}^{3} } \\ \\ : \implies \: \frac{1}{x} + \frac{3}{1} + \frac{4}{x} + \frac{2}{ {x}^{3} } \\ \\ : \implies \: {x}^{ - 1} + 3 + 4 {x}^{ - 1} + 2 {x}^{ - 3 } \\ \\ : \implies \: \frac{d( {x}^{ - 1} + 3 \: + 4 {x}^{ - 1} + 2 {x}^{ - 3}) }{dx} \\ \\ : \implies \frac{d( {x}^{ - 1} )}{dx} + \frac{d(3)}{dx} + \frac{d(4 {x}^{ - 1} )}{dx} + \frac{d(2 {x}^{ - 3}) }{dx} \\ \\ : \implies \: \frac{d( {x}^{ - 1}) }{dx} + \frac{d(3)}{dx} + 4 \frac{d( {x}^{ - 1} )}{dx} + 2 \frac{d( {x}^{ - 3} )}{dx} \\ \\ : \implies - 1x {}^{ - 1 - 1} + 0 + 4 \times - 1x {}^{ - 1 - 1} + 2 \times - 3x {}^{ - 3 - 1} \\ \\ : \implies - 1x {}^{ - 2} - 4x {}^{ - 2} - 6x {}^{ - 4} \\ \\ : \implies \frac{ - 1}{ {x}^{2} } - \frac{4}{ {x}^{2} } - \frac{6}{ {x}^{4} }$