Question

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

Answer 1

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