Question

Y=x^2×logx .Find the second order derivatives

Answer 1

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Answer 2

Answer:

Question:-

$\bf \:If \: y \: = {x}^{2} log(x) \: find \: \dfrac{ {d}^{2}y }{ {dx}^{2} }$

Formula used :-

$\bf \:\dfrac{d}{dx} {x}^{n} = n {x}^{n - 1}$

$\bf \:\dfrac{d}{dx} log(x) = \dfrac{1}{x}$

$\bf \:\dfrac{d}{dx}u.v \: = u \dfrac{dv}{dx} + v \dfrac{du}{dx}$

Solution :-

$\bf \:y \: = {x}^{2} log(x)$

Differnatiate w. r. t. x

$\bf\implies \:\dfrac{d}{dx}y = \dfrac{d}{dx} {x}^{2} log(x)$

$\bf\implies \:\dfrac{dy}{dx} = x²\dfrac{d}{dx} log(x) + log(x) \dfrac{d}{dx}x²$

$\bf\implies \:\dfrac{dy}{dx} = {x}^{2} \times \dfrac{1}{x} + log(x) \times 2x$

$\bf\implies \:\dfrac{dy}{dx} = x + 2x log(x)$

$\bf\implies \:\dfrac{dy}{dx} = x(1 + 2 logx )$

Differnatiate w. r. t. x

$\bf\implies \:\dfrac{ {d}^{2}y }{ {dx}^{2} } = x\dfrac{d}{dx}(1 + 2logx) + (1 + 2logx)\dfrac{d}{dx}x$

$\bf\implies \:\dfrac{ {d}^{2}y }{ {dx}^{2} } =x \times 2 \times \dfrac{1}{x} + (1 + 2logx) \times 1$

$\bf\implies \:\dfrac{ {d}^{2}y }{ {dx}^{2} } =2 + 1 + 2logx$

$\bf\implies \:\dfrac{ {d}^{2}y }{ {dx}^{2} } =3 + 2logx$

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