Question

$\log \left(\frac{1}{\sqrt{x}}\right)+5x^a-3a^x+\sqrt[3]{x^2}+6\sqrt[4]{x^{-3}}$

Answer 1

$$\begin{gathered} \frac{d}{dx}\left(\log \left(\frac{1}{\sqrt{x}}\right)+5x^a-3a^x+\sqrt[3]{x^2}+6\sqrt[4]{x^{-3}}\right) \\ =\frac{d}{dx}\left[l\mathrm{og}\left(x^{\frac{-1}{2}}\right)\right]+5\frac{d}{dx}\left(x^a\right)-3\frac{d}{dx}\left(a^x\right)+\frac{d}{dx}\left(x^{\frac{2}{3}}\right)+6\frac{d}{dx}\left(x^{\frac{-3}{4}}\right) \\ =\frac{d}{dx}\left(\frac{-1}{2}\log x\right)+5\frac{d}{dx}\left(x^a\right)-3\frac{d}{dx}\left(a^x\right)+\frac{d}{dx}\left(x^{\frac{2}{3}}\right)+6\frac{d}{dx}\left(x^{\frac{-3}{4}}\right) \\ =\frac{-1}{2}.\frac{1}{x}+5a{x}^{a-1}-3a^x\log a+\frac{2}{3}{x}^{\frac{-1}{3}}+6\left(\frac{-3}{4}\right)x^{\frac{-7}{4}} \\ =\frac{-1}{2x}+5a{x}^{a-1}-3a^x\log a+\frac{2}{3}{x}^{\frac{-1}{3}}-\frac{9}{2}{x}^{\frac{-7}{4}} \end{gathered}$$