Question

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Answer 1

$1)\; \; y=6x^{-1}-\frac{5}{\sqrt[3]{x}}+2\sqrt{x}\\\\y'=-6x^{-2}-5\cdot (-\frac{1}{3})x^{-\frac{4}{3}}+\frac{2}{2\sqrt{x}}=-\frac{6}{x^2}+\frac{5}{3\sqrt[3]{x^4}}+\frac{1}{\sqrt{x}}\\\\2)\; \; y=(tgx+5)(2^{x}-3)\\\\y'=\frac{1}{cos^2x}(2^{x}-3)+(tgx+5)\cdot 2^{x}\, ln2\\\\3)\; \; y=\frac{\sqrt{x}+1}{x+5}\\\\y'=\frac{\frac{1}{2\sqrt{x}}(x+5)-(\sqrt{x}+1)\cdot 1}{(x+5)^2}=\frac{x+5-2x-2\sqrt{x}}{2\sqrt{x}\cdot (x+5)^2}=\frac{5-x-2\sqrt{x}}{2\sqrt{x}\cdot (x+5)^2}\\\\4)\; \; y=sin^32x\\\\y'=3sin^22x\cdot (sin2x)'=3sin^22x\cdot cos2x\cdot 2=6sin^22x\cdot cos2x$

$5)\; \; y=x^3+e^{3x}\\\\y'=3x^2+3e^{3x}\\\\6)\; \; y=5\, ln^2(x+2)\\\\y'=5\cdot 2\cdot ln(x+2)\cdot (ln(x+2))'=10\cdot ln(x+2)\cdot \frac{1}{x+2}$