Các quy tắc tính đạo hàm

$y = {x^4} - 3{x^2} + 2x - 1 \Rightarrow y' = 4{x^3} - 3.2x + 2 = 4{x^3} - 6x + 2$

$y' = \dfrac{{\left( {2x + 1} \right)'.\left( {x + 2} \right) - \left( {2x + 1} \right)\left( {x + 2} \right)'}}{{{{\left( {x + 2} \right)}^2}}} = \dfrac{{2\left( {x + 2} \right) - 2x - 1}}{{{{\left( {x + 2} \right)}^2}}} = \dfrac{3}{{{{\left( {x + 2} \right)}^2}}}$

$f\left( x \right) = \sqrt[3]{x} = {x^{\frac{1}{3}}} \Rightarrow f'\left( x \right) = \dfrac{1}{3}.{x^{\frac{1}{3} - 1}} = \dfrac{1}{3}{x^{ - \frac{2}{3}}} = \dfrac{1}{3}\dfrac{1}{{{x^{\frac{2}{3}}}}} = \dfrac{1}{3}\dfrac{1}{{\sqrt[3]{{{x^2}}}}}$ $\Rightarrow f'\left( 8 \right) = \dfrac{1}{3}.\dfrac{1}{{\sqrt[3]{{{8^2}}}}} = \dfrac{1}{{12}}$

Bước 1:

$y' = \dfrac{{3'\left( {1 - x} \right) - 3\left( {1 - x} \right)'}}{{{{\left( {1 - x} \right)}^2}}} = \dfrac{{ - 3.\left( { - 1} \right)}}{{{{\left( {1 - x} \right)}^2}}}$$= \dfrac{3}{{{{\left( {1 - x} \right)}^2}}}$

Bước 2:

Ta có $y'=\dfrac{3}{{{{\left( {1 - x} \right)}^2}}} > 0\,\,\forall x \ne 1$

$\Rightarrow$Tập nghiệm của bất phương trình $y' < 0$ là $\emptyset$.

Đáp án A: $y' = \dfrac{{\left( {{x^3} + 1} \right)'.x - \left( {{x^3} + 1} \right)x'}}{{{x^2}}}$ $= \dfrac{{3{x^2}.x - {x^3} - 1}}{{{x^2}}} = \dfrac{{2{x^3} - 1}}{{{x^2}}}$

Đáp án B:

$\begin{array}{l}y = \dfrac{{3\left( {x + 1} \right)}}{{{x^2}}}\\ \Rightarrow y' = 3.\dfrac{{\left( {x + 1} \right)'.{x^2} - \left( {x + 1} \right)\left( {{x^2}} \right)'}}{{{x^4}}}\\ = 3\dfrac{{{x^2} - 2x\left( {x + 1} \right)}}{{{x^4}}}\\ = 3\dfrac{{ - {x^2} - 2x}}{{{x^4}}} =  - 3\dfrac{{x + 2}}{{{x^3}}}\end{array}$

Đáp án C: $y' = \dfrac{{\left( {{x^3} + 5x - 1} \right)'.x - \left( {{x^3} + 5x - 1} \right).x'}}{{{x^2}}}$ $= \dfrac{{\left( {3{x^2} + 5} \right).x - {x^3} - 5x + 1}}{{{x^2}}}$ $= \dfrac{{2{x^3} + 1}}{{{x^2}}} = 2x + \dfrac{1}{{{x^2}}}$

$\begin{array}{l}y = \dfrac{1}{{{x^3}}} - \dfrac{1}{{{x^2}}} = {x^{ - 3}} - {x^{ - 2}}\\ \Rightarrow y' =  - 3{x^{ - 4}} - \left( { - 2} \right){x^{ - 3}} = \dfrac{{ - 3}}{{{x^4}}} + \dfrac{2}{{{x^3}}}\end{array}$

$\begin{array}{l}y' = \dfrac{{\left( {ax + b} \right)'\left( {cx + d} \right) - \left( {ax + b} \right)\left( {cx + d} \right)'}}{{{{\left( {cx + d} \right)}^2}}} = \dfrac{{a\left( {cx + d} \right) - c\left( {ax + b} \right)}}{{{{\left( {cx + d} \right)}^2}}}\\ = \dfrac{{acx + ad - acx - bc}}{{{{\left( {cx + d} \right)}^2}}} = \dfrac{{ad - bc}}{{{{\left( {cx + d} \right)}^2}}}\end{array}$

$\begin{array}{l}y' = \dfrac{{\left( {{x^2} - x + 1} \right)'\left( {x - 1} \right) - \left( {{x^2} - x + 1} \right)\left( {x - 1} \right)'}}{{{{\left( {x - 1} \right)}^2}}} = \dfrac{{\left( {2x - 1} \right)\left( {x - 1} \right) - \left( {{x^2} - x + 1} \right)}}{{{{\left( {x - 1} \right)}^2}}}\\ = \dfrac{{2{x^2} - 2x - x + 1 - {x^2} + x - 1}}{{{{\left( {x - 1} \right)}^2}}} = \dfrac{{{x^2} - 2x}}{{{{\left( {x - 1} \right)}^2}}}\end{array}$

$\begin{array}{l}y = {\left( {{x^7} + x} \right)^2} = {x^{14}} + 2{x^8} + {x^2}\\ \Rightarrow y' = 14{x^{13}} + 2.8{x^7} + 2x\\ = 14{x^{13}} + 16{x^7} + 2x\\ = 2\left( {7{x^{13}} + 8{x^7} + x} \right)\\ = 2\left( {7{x^{13}} + 7{x^7} + {x^7} + x} \right) \\= 2\left[ {7{x^6}\left( {{x^7} + x} \right) + {x^7} + x} \right]\\= 2\left( {{x^7} + x} \right)\left( {7{x^6} + 1} \right)\end{array}$

$\begin{array}{l}y = \dfrac{1}{{x\sqrt x }} = \dfrac{1}{{x.{x^{\frac{1}{2}}}}} = \dfrac{1}{{{x^{1 + \frac{1}{2}}}}} = \dfrac{1}{{{x^{\frac{3}{2}}}}} = {x^{ - \frac{3}{2}}}\\ \Rightarrow y' =  - \dfrac{3}{2}{x^{ - \frac{3}{2} - 1}} =  - \dfrac{3}{2}{x^{ - \frac{5}{2}}} =  - \dfrac{3}{2}\dfrac{1}{{{x^{\frac{5}{2}}}}} =  - \dfrac{3}{2}\dfrac{1}{{{x^2}\sqrt x }}\end{array}$

Bước 1:

$\begin{array}{l}y = \sin 2x = 2\sin x\cos x\\ \Rightarrow y' = \left( {2\sin x\cos x} \right)'\\ = 2\left( {\sin x\cos x} \right)'\\ = 2\left[ {\left( {\sin x} \right)'.\cos x + \sin x.\left( {\cos x} \right)'} \right]\end{array}$

Bước 2:

$= 2\left( {{{\cos }^2}x - {{\sin }^2}x} \right)\\ = 2\cos 2x$

$\begin{array}{l}y' = \dfrac{{\left( {2{x^2} + 3x - 1} \right)'\left( {{x^2} - 5x + 2} \right) - \left( {2{x^2} + 3x - 1} \right)\left( {{x^2} - 5x + 2} \right)'}}{{{{\left( {{x^2} - 5x + 2} \right)}^2}}}\\y' = \dfrac{{\left( {4x + 3} \right)\left( {{x^2} - 5x + 2} \right) - \left( {2{x^2} + 3x - 1} \right)\left( {2x - 5} \right)}}{{{{\left( {{x^2} - 5x + 2} \right)}^2}}}\\y' = \dfrac{{4{x^3} - 20{x^2} + 8x + 3{x^2} - 15x + 6 - 4{x^3} - 6{x^2} + 2x + 10{x^2} + 15x - 5}}{{{{\left( {{x^2} - 5x + 2} \right)}^2}}}\\y' = \dfrac{{ - 13{x^2} + 10x + 1}}{{{{\left( {{x^2} - 5x + 2} \right)}^2}}}\end{array}$

Có: $f'\left( x \right) = 3{x^2} - 3.2x = 3{x^2} - 6x$

$f'\left( x \right) < 0 \Leftrightarrow 3{x^2} - 6x < 0 \Leftrightarrow 0 < x < 2$

$\begin{array}{l}f\left( x \right) = {\left( {\sqrt x  - \dfrac{1}{{\sqrt x }}} \right)^3} = {\left( {\sqrt x } \right)^3} - 3{\left( {\sqrt x } \right)^2}.\dfrac{1}{{\sqrt x }} + 3\sqrt x {\left( {\dfrac{1}{{\sqrt x }}} \right)^2} - {\left( {\dfrac{1}{{\sqrt x }}} \right)^3}\\f\left( x \right) = {x^{\dfrac{3}{2}}} - 3\sqrt x  + \dfrac{3}{{\sqrt x }} - \dfrac{1}{{{x^{\dfrac{3}{2}}}}}\\f\left( x \right) = {x^{\dfrac{3}{2}}} - 3\sqrt x  + 3{x^{ - \dfrac{1}{2}}} - {x^{ - \dfrac{3}{2}}}\\f'\left( x \right) = \dfrac{3}{2}{x^{\dfrac{3}{2} - 1}} - \dfrac{3}{{2\sqrt x }} + 3.\left( { - \dfrac{1}{2}} \right){x^{ - \dfrac{1}{2} - 1}} + \dfrac{3}{2}{x^{ - \dfrac{3}{2} - 1}}\\f'\left( x \right) = \dfrac{3}{2}\sqrt x  - \dfrac{3}{{2\sqrt x }} - \dfrac{3}{2}{x^{ - \dfrac{3}{2}}} + \dfrac{3}{2}{x^{ - \dfrac{5}{2}}}\\f'\left( x \right) = \dfrac{3}{2}\left( {\sqrt x  - \dfrac{1}{{\sqrt x }} - \dfrac{1}{{x\sqrt x }} + \dfrac{1}{{{x^2}\sqrt x }}} \right)\end{array}$

$\begin{array}{l}y = {\tan ^2}x - co{t^2}x = \left( {\tan x - \cot x} \right)\left( {\tan x + \cot x} \right)\\y' = \left( {\tan x - \cot x} \right)'\left( {\tan x + \cot x} \right) + \left( {\tan x - \cot x} \right)\left( {\tan x + \cot x} \right)'\\y' = \left( {\dfrac{1}{{{{\cos }^2}x}} + \dfrac{1}{{{{\sin }^2}x}}} \right)\left( {\tan x + \cot x} \right) + \left( {\tan x - \cot x} \right)\left( {\dfrac{1}{{{{\cos }^2}x}} - \dfrac{1}{{{{\sin }^2}x}}} \right)\\y' = \dfrac{{\tan x}}{{{{\cos }^2}x}} + \dfrac{{\cot x}}{{{{\cos }^2}x}} + \dfrac{{\tan x}}{{{{\sin }^2}x}} + \dfrac{{\cot x}}{{{{\sin }^2}x}} + \dfrac{{\tan x}}{{{{\cos }^2}x}} - \dfrac{{\tan x}}{{{{\sin }^2}x}} - \dfrac{{\cot x}}{{{{\cos }^2}x}} + \dfrac{{\cot x}}{{{{\sin }^2}x}}\\y' = 2\dfrac{{\tan x}}{{{{\cos }^2}x}} + 2\dfrac{{\cot x}}{{{{\sin }^2}x}}\end{array}$

$\begin{array}{l}f\left( x \right) = \tan \left( {x - \dfrac{{2\pi }}{3}} \right) = \dfrac{{\tan x - \tan \dfrac{{2\pi }}{3}}}{{1 + \tan x.\tan \dfrac{{2\pi }}{3}}} = \dfrac{{\tan x + \sqrt 3 }}{{1 - \sqrt 3 \tan x}}\\f'\left( x \right) = \dfrac{{\left( {\tan x + \sqrt 3 } \right)'\left( {1 - \sqrt 3 \tan x} \right) - \left( {\tan x + \sqrt 3 } \right)\left( {1 - \sqrt 3 \tan x} \right)'}}{{{{\left( {1 - \sqrt 3 \tan x} \right)}^2}}}\\f'\left( x \right) = \dfrac{{\dfrac{1}{{{{\cos }^2}x}}\left( {1 - \sqrt 3 \tan x} \right) - \left( {\tan x + \sqrt 3 } \right)\left( { - \dfrac{{\sqrt 3 }}{{{{\cos }^2}x}}} \right)}}{{{{\left( {1 - \sqrt 3 \tan x} \right)}^2}}}\\f'\left( x \right) = \dfrac{{\dfrac{1}{{{{\cos }^2}x}} - \dfrac{{\sqrt 3 \tan x}}{{{{\cos }^2}x}} + \dfrac{{\sqrt 3 \tan x}}{{{{\cos }^2}x}} + \dfrac{3}{{{{\cos }^2}x}}}}{{{{\left( {1 - \sqrt 3 \tan x} \right)}^2}}}\\f'\left( x \right) = \dfrac{4}{{{{\cos }^2}x{{\left( {1 - \sqrt 3 \tan x} \right)}^2}}}\\ \Rightarrow f'\left( 0 \right) = \dfrac{4}{{1\left( {1 - \sqrt 3 .0} \right)}} = 4\end{array}$

Bước 1:

$\begin{array}{l} \left( {{{\tan }^2}\dfrac{x}{2}} \right)' = 2\tan \dfrac{x}{2}\left( {\tan \dfrac{x}{2}} \right)'\\  \end{array}$

Bước 2:

$= 2\tan \dfrac{x}{2}.\dfrac{{\left( {\dfrac{x}{2}} \right)'}}{{{{\cos }^2}\dfrac{x}{2}}}$

$= 2\tan \dfrac{x}{2}.\dfrac{\dfrac{1}{2}}{{{{\cos }^2}\dfrac{x}{2}}}\\ = \dfrac{{\sin \dfrac{x}{2}}}{{\cos \dfrac{x}{2}}}.\dfrac{1}{{{{\cos }^2}\dfrac{x}{2}}} = \dfrac{{\sin \dfrac{x}{2}}}{{{{\cos }^3}\dfrac{x}{2}}}$

$\begin{array}{l}y = x\left( {2x - 1} \right)\left( {3x + 2} \right)\left( {\sin x - \cos x} \right)' = \left( {6{x^3} + {x^2} - 2x} \right)\left( {\sin x + \cos x} \right)\\ \Rightarrow y' = \left( {6{x^3} + {x^2} - 2x} \right)'\left( {\sin x + \cos x} \right) + \left( {6{x^3} + {x^2} - 2x} \right)\left( {\sin x + \cos x} \right)'\\y' = \left( {18{x^2} + 2x - 2} \right)\left( {\sin x + \cos x} \right) + \left( {6{x^3} + {x^2} - 2x} \right)\left( {\cos x - \sin x} \right)\\y' = \sin x\left( {18{x^2} + 2x - 2 - 6{x^3} - {x^2} + 2x} \right) + \cos x\left( {18{x^2} + 2x - 2 + 6{x^3} + {x^2} - 2x} \right)\\y' = \sin x\left( { - 6{x^3} + 17{x^2} + 4x - 2} \right) + \cos x\left( {6{x^3} + 19{x^2} - 2} \right)\end{array}$

Với $x > 1$ ta có: $f\left( x \right) = {x^2} - 3x + 1 \Rightarrow f'\left( x \right) = 2x - 3$

Với $x < 1$ ta có : $f\left( x \right) = 2x + 2 \Leftrightarrow f'\left( x \right) = 2$

Với $x = 1$ ta có : $\mathop {\lim }\limits_{x \to {1^ + }} f\left( x \right) = \mathop {\lim }\limits_{x \to {1^ + }} \left( {{x^2} - 3x + 1} \right) =  - 1 \ne f\left( 1 \right) = 4$ $\Rightarrow$ Hàm số không liên tục tại $x = 1,$ do đó không có đạo hàm tại $x = 1.$

Vậy $f'\left( x \right) = \left\{ \begin{array}{l}2x - 3\,\,\,khi\,\,x > 1\\2\,\,\,\,\,\,\,\,\,\,\,\,\,khi\,\,x < 1\end{array} \right.$

$\begin{array}{l}y = \dfrac{{m{x^3}}}{3} - m{x^2} + \left( {3m - 1} \right)x + 1\\ \Rightarrow y' = m{x^2} - 2mx + 3m - 1\\y' \le 0,\forall x \in R \Rightarrow m{x^2} - 2mx + 3m - 1 \le 0,\forall x \in R\end{array}$

TH1: $m = 0,$ khi đó $BPT \Leftrightarrow  - 1 \le 0$ , đúng $\forall x \in R$

TH2: $\begin{array}{l}m \ne 0 \Leftrightarrow y' \le 0\,\,\forall x \in R \Leftrightarrow \left\{ \begin{array}{l}a = m < 0\\\Delta ' = {m^2} - m\left( {3m - 1} \right) \le 0\end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l}m < 0\\ - 2{m^2} + m \le 0\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}m < 0\\\left[ \begin{array}{l}m \le 0\\m \ge \dfrac{1}{2}\end{array} \right.\end{array} \right. \end{array}$ $\Leftrightarrow m < 0$

Kết hợp cả 2 trường hợp ta có $m \le 0$ là những giá trị cần tìm.