Differentiation of (√x+1/√x)² – InterviewSolution

1.

Differentiation of (√x+1/√x)²

Answer»

GIVEN :–

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• A FUNCTION $\rm{ \left( \sqrt{<klux>X</klux>} + \dfrac{1}{ \sqrt{x} } \right) }^{2}$

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TO FIND :–

• Differentiate form of given function = ?

SOLUTION :–

• LET the function –

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$\rm \implies y = { \left( \sqrt{x} + \dfrac{1}{ \sqrt{x} } \right) }^{2}$

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• Using identity –

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$\rm \implies { \left(a+b \right)}^{2} = {a}^{2} + {b}^{2} + 2ab$

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• So that –

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$\rm \implies y = { ( \sqrt{x})^{2} + \left( \dfrac{1}{ \sqrt{x} } \right) }^{2} + 2( \sqrt{x} ) \left( \dfrac{1}{ \sqrt{x} } \right)$

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$\rm \implies y = x+ \dfrac{1}{x} + 2$

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• We should write this as –

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$\rm \implies y = x+ {x}^{ - 1} + 2$

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• Using FORMULAS –

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$\rm \to \dfrac{d( {x}^{n})}{dx} = n {x}^{n - 1}$

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$\rm \to \dfrac{d(constant)}{dx} = 0$

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• Now differentiate with respect to 'x' –

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$\rm \implies \dfrac{dy}{dx} = \dfrac{d(x)}{dx} + \dfrac{d({x}^{ - 1})}{dx}+ \dfrac{d(2)}{dx}$

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$\rm \implies \dfrac{dy}{dx} = 1+( - 1){x}^{( - 1 - 1)}+0$

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$\rm \implies \dfrac{dy}{dx} = 1 - {x}^{ -2}$

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$\rm \implies \large{ \boxed{ \rm \dfrac{dy}{dx} = 1 - \dfrac{1}{{x}^{2}}}}$

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