Giải câu 1 bài Ôn tập chương 5: Đạo hàm

Giải câu 1 bài Ôn tập chương 5: Đạo hàm.

a) $y=\frac{x^3}{3}-\frac{x^2}{2}+x-5$

$y^{\prime }={(\frac{x^3}{3}-\frac{x^2}{2}+x-5)}^{\prime }$

$=\frac{3x^2}{3}-\frac{2x}{2}+1=x^2-x+1$

b) $y=\frac{2}{x}-\frac{4}{x^2}+\frac{5}{x^3}-\frac{6}{7x^4}$

$=2.\frac{1}{x}-4.\frac{1}{x^2}+5.\frac{1}{x^3}-\frac{6}{7}.\frac{1}{x^4}$

$=\frac{1}{x^4}(2x^3-4x^2+5x-\frac{6}{7})$

$y^{\prime }={\left(\frac{1}{x^4}\right)}^{\prime }.(2x^3-4x^2+5x-\frac{6}{7})+\left(\frac{1}{x^4}\right).{(2x^3-4x^2+5x-\frac{6}{7})}^{\prime }$

$=-\frac{4}{x^5}.(2x^3-4x^2+5x-\frac{6}{7})+\left(\frac{1}{x^4}\right).(6x^2-8x+5)$

$=-\frac{8}{x^2}+\frac{16}{x^3}-\frac{20}{x^4}+\frac{24}{7x^5}+\frac{6}{x^2}-\frac{8}{x^3}+\frac{5}{x^4}$

$=-\frac{2}{x^2}+\frac{8}{x^3}-\frac{15}{x^4}+\frac{24}{7x^5}$

c) $y=\frac{3x^2-6x+7}{4x}$

$y^{\prime }={\left(\frac{3x^2-6x+7}{4x}\right)}^{\prime }$

$=\frac{(3x^2-6x+7{)}^{\prime }4x-(4x{)}^{\prime }(3x^2-6x+7)}{16x^2}$

$=\frac{(6x-6)4x-4(3x^2-6x+7)}{16x^2}$

$=\frac{3x^2-7}{4x^2}$

d) $y=(\frac{2}{x}+3x)(\sqrt{x}-1)$

$y^{\prime }={\left[(\frac{2}{x}+3x)(\sqrt{x}-1)\right]}^{\prime }$

$=(-\frac{2}{x^2}+3)(\sqrt{x}-1)+\frac{1}{2\sqrt{x}}.(\frac{2}{x}+3x)$

$=-\frac{2\sqrt{x}}{x^2}+\frac{2}{x^2}+3\sqrt{x}-3+\frac{1}{x\sqrt{x}}+\frac{3x}{2\sqrt{x}}$

$=-\frac{4\sqrt{x}}{2x^2}+\frac{4}{2x^2}+\frac{12x^2\sqrt{x}}{2x^2}-\frac{6x^2}{2x^2}+\frac{2\sqrt{x}}{2x^2}+\frac{3x^2\sqrt{x}}{2x^2}$

$=\frac{9x^2\sqrt{x}-6x^2-2\sqrt{x}+4}{2x^2}$ 

e) $y=\frac{1+\sqrt{x}}{1-\sqrt{x}}$

$y^{\prime }={\left(\frac{1+\sqrt{x}}{1-\sqrt{x}}\right)}^{\prime }=\frac{\frac{1}{2\sqrt{x}}(1-\sqrt{x})+\frac{1}{2\sqrt{x}}(1+\sqrt{x})}{{(1-\sqrt{x})}^2}$

$=\frac{1-\sqrt{x}+1+\sqrt{x}}{2\sqrt{x}(1-\sqrt{x}{)}^2}$

$=\frac{1}{\sqrt{x}(1-\sqrt{x}{)}^2}$

f) $y=\frac{-x^2+7x+5}{x^2-3x}$

$y^{\prime }={\left(\frac{-x^2+7x+5}{x^2-3x}\right)}^{\prime }$

$=\frac{(-2x+7)(x^2-3x)-(2x-3)(-x^2+7x+5)}{{(x^2-3x)}^2}$

$=\frac{-2x^3+6x^2+7x^2-21x-(-2x^3+14x^2+10x+3x^2-21x-15)}{(x^2-3x{)}^2}$

$=\frac{-4x^2-10x+15}{{(x^2-3x)}^2}$