Question

Differentiate the following functions with respect to x :

$\frac{2^{x}cotx}{\sqrt{x}}$

Answer 1

We have,

$\frac{d}{dx}\left(\frac{2^{x}cotx}{\sqrt{x}}\right)$

Using quotient rule, we get

$\frac{\sqrt{x}\frac{d}{dx}\left(2^{x}cotx\right)-\left(2^{x}cotx\right)\frac{d}{dx}\left(\sqrt{x}\right)}{(\sqrt{x}{)}^2}$

$=\frac{\sqrt{x}(2^x\frac{d}{dx}cot+cotx\frac{d}{dx}{2}^x)-2^{x}cotx\times \frac{1}{2}{x}^{\frac{-1}{2}}}{(\sqrt{x}{)}^2}$

$=\frac{\sqrt{x}(2^x-cose{c}^{2}x+cotx\times log2\times 2^x)-2^{x}cotx\times \frac{1}{2\sqrt{x}}}{(\sqrt{x}{)}^2}$

$=\frac{2^x(-xcose{c}^{2}x+xcotx\times logx-\left(\frac{1}{2}cotx\right))}{(\sqrt{x}{)}^2\times \sqrt{x}}=\frac{2^x(-xcose{c}^{2}x+xcotx\times log2-\left(\frac{1}{2}\right)cotx)}{x^{\frac{3}{2}}}$