Question

Differentiate with respect to 'x': i)x(x+5)/(x-1)(x+2). Please solve it​

Answer 1

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

$\dashrightarrow\:\:\sf \dfrac{x(x + 5)}{(x - 1)(x + 2)}$

$\dashrightarrow\:\:\sf \dfrac{ {x}^{2} + 5x}{ {x}^{2} + 2x - x - 2}$

$\dashrightarrow\:\:\sf \dfrac{ {x}^{2} + 5x}{ {x}^{2} + x - 2}$

$\dag \: \underline{\textbf{By using Division Rule :}}$

$\dashrightarrow\:\:\sf \dfrac{dy}{dx} = \dfrac{d}{dx} \bigg( \dfrac{u}{v} \bigg) = \dfrac{(v) \dfrac{d}{dx}(u) - (u) \dfrac{d}{dx}(v)}{ {v}^{2} }$

$\dashrightarrow\:\:\sf \dfrac{d}{dx} \bigg( \dfrac{ {x}^{2} + 5x}{ {x}^{2} + x - 2} \bigg)$

$\dashrightarrow\:\:\sf \dfrac{dy}{dx} = \dfrac{d}{dx} \bigg( \dfrac{ {x}^{2} + 5x}{ {x}^{2} + x - 2} \bigg) = \dfrac{({x}^{2} + x - 2) \dfrac{d}{dx}({x}^{2} + 5x) - ({x}^{2} + 5x) \dfrac{d}{dx}({x}^{2} + x - 2)}{ {( {x}^{2} + x - 2) }^{2} }$

$\dashrightarrow\:\:\sf \dfrac{( {x}^{2} + x - 2)(2x + 5) - ( {x}^{2} + 5x)(2x + 1)}{( {x}^{2} + x - 2)^{2} }$

$\dashrightarrow\:\:\sf \dfrac{( {2x}^{3} + {2x}^{2} - 4x + {5x}^{2} + 5x - 10)- ( {2x}^{3} + {10x}^{2} + {x}^{2} + 5x)}{( {x}^{2} + x - 2)^{2} }$

$\dashrightarrow\:\:\sf \dfrac{( {2x}^{3} + {7x}^{2} + x - 10)- ( {2x}^{3} + {11x}^{2} + 5x)}{( {x}^{2} + x - 2)^{2} }$

$\dashrightarrow\:\:\sf \dfrac{ {2x}^{3} + {7x}^{2} + x - 10- {2x}^{3} - {11x}^{2} - 5x}{( {x}^{2} + x - 2)^{2} }$

$\dashrightarrow\:\: \underline{ \boxed{\frak{\dfrac{ { - 4x}^{2} - 4 x - 10}{( {x}^{2} + x - 2)^{2} }}}}$

Answer 2

${\bf{\underline{\underline{ANSWER\:IN\:THE\:ATTACHMENT}}}}$

Thank You !