Question

find the derivative of given function w.r.t independent variable 't' :
f(t)= t²-1/ t²+t-2​

Answer 1

$\dag\:\underline{\sf AnsWer :}$

$:\implies\sf f(t) = \dfrac{t^2 - 1}{t^2 + t - 2}$

$:\implies\sf f(t) = \dfrac{(t - 1)(t + 1)}{t^2 + 2t - t - 2}$

$:\implies\sf f(t) = \dfrac{(t - 1)(t + 1)}{t(t + 2) -1( t + 2)}$

$:\implies\sf f(t) = \dfrac{(t - 1)(t + 1)}{(t - 1)( t + 2)}$

$:\implies\sf f(t) = \dfrac{ \cancel{(t - 1)}(t + 1)}{ \cancel{(t - 1)}( t + 2)}$

$:\implies\sf f(t) = \dfrac{(t + 1)}{ (t + 2)}$

$\bigstar\:\underline{\textbf{ By using division rule : }}$

$:\implies\sf \dfrac{d}{dt} \dfrac{(t + 1)}{ (t + 2)}$

$:\implies\sf \dfrac{(t + 2) \dfrac{d}{dt}(t + 1) - (t + 1)\dfrac{d}{dt}(t + 2)}{ (t + 2) ^{2} }$

$:\implies\sf \dfrac{(t + 2)(1) - (t + 1)(1)}{ (t + 2) ^{2} }$

$:\implies\sf \dfrac{t + 2- t - 1}{ (t + 2) ^{2} }$

$:\implies\sf \dfrac{ 2 - 1}{ (t + 2) ^{2} }$

$:\implies \underline{ \boxed{\frak{ \dfrac{1}{ (t + 2) ^{2} }}}}$

Answer 2

${\boxed{\underline{ \orange{\tt{Required \: \: answer:-}}}}}$

${ \boxed{ \underline{ \huge { \sf{ \bf{\frac{1}{ {(t +2)}^{2} } }}}}}}$

★USING:-

• Factorisation

★SOLUTION:-

$: \implies \sf \large{ \frac{d}{dx} ( \frac{ {t}^{2} - 1 }{ {t}^{2} + t - 2} )}$

$: \implies \sf \large{ \frac{d}{dx} ( \frac{(t + 1)(t - 1)}{t(t + 2) - 1(t + 2)} )}$

$: \implies \sf \large{ \frac{d}{dx} ( \frac{(t + 1) \cancel{(t - 1)}}{ \cancel{(t - 1)}(t + 2)} ) }$

$: \implies \sf \large{ \frac{d}{dx}( \frac{t + 1}{t + 2} )}$

$: \implies \sf \large{ \frac{(t + 2) \frac{d}{dx}(t + 1) - (t + 1) \frac{d}{dx} (t + 2) }{ {(t + 2)}^{2} } }$

$: \implies \sf \large{ \frac{(t + 2) \times 1 - (t + 1) \times 1}{ {(t + 2)}^{2} } }$

$: \implies \sf \large{ \frac{t + 2 - t - 1}{ {(t + 2)}^{2} } }$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: { \boxed{ \underline{ \purple{ \large{ \sf{ \bf{ \frac{1}{ {(t + 2)}^{2} }}}}}}}}$