Question

differentiate the following question.
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Answer 1

x√y + y√8 = 1

Differentiating on both sides with respect to x

$\implies \dfrac{d}{dx} (x \sqrt{y} + y \sqrt{8} ) = \dfrac{d}{dx} (1)$

Using sum rule d( u + v ) / dx = du/dx + dv/dx and constant rule d( k) / dx = 0

$\implies \dfrac{d}{dx} (x \sqrt{y} )+ \dfrac{d}{dx} ( y \sqrt{8} ) = 0$

Using product rule d(uv)/ dx = u × dv/dx + v × du/dx

$\implies x.\dfrac{d}{dx} (\sqrt{y} ) + \sqrt{y} . \dfrac{d}{dx}(x) + \dfrac{d}{dx} ( y \sqrt{8} ) = 0$

Using chain rule d/dx { f( g(x) ) } = f'( g(x) ) × g'( x) and Power rule d/dx ( x^n ) = nx^( n - 1 )

$\implies x \bigg( \dfrac{1}{2 \sqrt{y} } \bigg) \dfrac{dy}{dx} + \sqrt{y} .1 {x}^{1 - 1} + \dfrac{d}{dx} ( y \sqrt{8} ) = 0$

$\implies \bigg(\dfrac{x}{2 \sqrt{y} } + \sqrt{8} \bigg) \dfrac{dy}{dx} = - \sqrt{y}$

$\implies \boxed{\dfrac{dy}{dx} = - \dfrac{2y}{x + 2 \sqrt{8y} } }$