Question

differentiate the following function (x-1/x)^2​

Answer 1

We have to differentiate the given function :-

$\rm y = {\bigg (x - \dfrac{1}{x} \bigg)}^{2}$

$\rm \longrightarrow \dfrac{dy}{dx} = \dfrac{d{\bigg (x - \dfrac{1}{x} \bigg)}^{2} }{dx} \\ \\ \\ \rm \longrightarrow \dfrac{dy}{dx} = 2 \times \bigg(x - \dfrac{1}{x} \bigg) \times \dfrac{d{\bigg (x - \dfrac{1}{x} \bigg)} }{dx}\\ \\ \\ \rm \longrightarrow \dfrac{dy}{dx} = 2 \times \bigg(x - \dfrac{1}{x} \bigg) \times \bigg ( \dfrac{d{ (x )} }{dx} - \dfrac{d{ ( \frac{1}{x} )} }{dx}\bigg)\\ \\ \\ \rm \longrightarrow \dfrac{dy}{dx} = 2 \times \bigg(x - \dfrac{1}{x} \bigg) \times \bigg (1- \dfrac{d{ ( {x}^{{ - 1}} )} }{dx}\bigg)\\ \\ \\ \rm \longrightarrow \dfrac{dy}{dx} = 2 \times \bigg(x - \dfrac{1}{x} \bigg) \times \bigg (1- {( - {x}^{ - 2} ) }\bigg)\\ \\ \\ \rm \longrightarrow \dfrac{dy}{dx} = 2 \times \bigg(x - \dfrac{1}{x} \bigg) \times \bigg (1 + {x}^{ - 2}\bigg)\\ \\ \\ \rm \longrightarrow \dfrac{dy}{dx} = 2 \times \bigg(x - \dfrac{1}{x} \bigg) \times \bigg (1 + \dfrac{1}{ {x}^{2} }\bigg)\\ \\ \\ \rm \longrightarrow \dfrac{dy}{dx} = 2 \times \bigg(x + \dfrac{1}{x} - \dfrac{1}{x} -\dfrac{1}{ {x}^{3} } \bigg)\\ \\ \\ \rm \longrightarrow \dfrac{dy}{dx} = 2 \times \bigg(x -\dfrac{1}{ {x}^{3} } \bigg)\\ \\ \\ \rm \longrightarrow \dfrac{dy}{dx} = \dfrac{ 2{x}^{4} - 2}{ {x}^{3} }$