Question

Differentiate the function w.r.t.x:
$\rm \sqrt{x}.\cot x$

Answer 1

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Answer 2

We know that

$\frac{d {x}^{n} }{dx} = n {x}^{n - 1} \\ \\ \frac{d \: cot \: x}{dx} = -cosec^{2} \: x$
So, to differentiate given function with respect to x, we have to apply UV formula of differentiation.

$\frac{d(UV)}{dx} = U \frac{dV}{dx} + V\frac{dU}{dx} \\ \\ here \: U = \sqrt{x} \\ \\ V = cot \: x \\ \\ \frac{d( \sqrt{x} \: cot \: x)}{dx} = \sqrt{x} \bigg(\frac{d \: cot \: x}{dx}\bigg) + cot \: x\bigg(\frac{d \sqrt{x} }{dx}\bigg) \\ \\\frac{d( \sqrt{x} \: cot \: x)}{dx} = - \sqrt{x} \: {cosec}^{2} x + \frac{1}{2 \sqrt{x} } cot \: x$
Hope it helps you