Question

Differentiate the function w.r.t.x:
x². sin x . log x

Answer 1

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Answer 2

We know that

$\frac{d {x}^{n} }{dx} = n {x}^{n - 1} \\ \\ \frac{d \: sin\: x}{dx} = cos\: x \\ \\\frac{d \:log x}{dx} = \frac{1}{x}$
So, to differentiate given function with respect to x, we have to apply UV formula of differentiation.

$\frac{d(UV)}{dx} = U \frac{dV}{dx} + V\frac{dU}{dx} \\ \\ here \: U = (x^{2}\: log\:x) \\ \\ V = sin \: x\\ \\ \frac{d[(x^{2}\: log\:x)sin \: x)]}{dx} =\\\\ (x^{2}\: log\:x) \bigg(\frac{d \:sin \: x)}{dx}\bigg) + (sin \: x)\bigg(\frac{d (x^{2}\: log\:x) }{dx}\bigg) \\ \\\frac{d[(x^{2}\: log\:x)sin \: x)]}{dx}\\\\ = (x^{2}\: log\:x)( cos\: x)+(sin \: x)[ x^{2}\bigg(\frac{d\:log\: x}{dx} \bigg)+ log\:x\bigg(\frac{dx^{2}}{dx}\bigg) \\ \\= (x^{2}\: log\:x)( cos\: x)+(sin \: x)[ x^{2}\frac{1}{x} + 2xlog\:x]\\\\=(x^{2}\: log\:x)( cos\: x)+(sin \: x)( x+ 2xlog\:x)$
Hope it helps you