Bài 1: Tìm đạo hàm của các hàm số sau:
a) (y = frac{x-1}{5x-2});
b) (y = frac{2x+3}{7-3x});
c) (y = frac{x^{2}+2x+3}{3-4x});
d) (y = frac{x^{2}+7x+3}{x^{2}-3x}).
a) ( y’=frac{left ( x-1 right )’.left ( 5x-2 right )-left ( x-1 right ).left ( 5x-2 right )’}{left ( 5x-2 right )^{2}}) = ( frac{5x-2-left ( x-1 right ).5}{left ( 5x-2 right )^{2}}) = ( frac{3}{left ( 5x-2 right )^{2}}).
b) ( y’=frac{left ( 2x+3 right )’.left ( 7-3x right )-left ( 2x+3 right ).left ( 7-3x right )’}{left ( 7-3x right )^{2}}) = ( frac{2left ( 7-3x right )-left ( 2x+3 right ).left ( -3 right )}{left ( 7-3x right )^{2}}) = ( frac{23}{left ( 7-3x right )^{2}}).
c) ( y’=frac{left ( x^{2}+2x+3 right )’.left ( 3-4x right )-left ( x^{2} +2x+3right ).left ( 3-4x right )’}{left ( 3-4x right )^{2}}) = ( frac{left ( 2x+2 right ).left ( 3-4x right )-left ( x^{2}+2x+3 right ).(-4)}{(3-4x)^{2}}) = ( frac{-2(2x^{2}-3x-9)}{(3-4x)^{2}}).
d) ( y’=frac{(x^{2}+7x+3)’.(x^{2}-3x)-(x^{2}+7x+3).(x^{2}-3x)’}{(x^{2}-3x)^{2}}) = ( frac{(2x-7).(x^{2}-3x)-(x^{2}+7x+3).(2x-3)}{(x^{2}-3x)^{2}}) = ( frac{-10x^{2}-6x+9}{(x^{2}-3x)^{2}}).
Bài 2: Giải các bất phương trình sau:
a) (y'<0) với ({{{x^2} + x + 2} over {x – 1}})
b) (y’≥0) với (y = frac{x^{2}+3}{x+1});
c) (y’>0) với (y = frac{2x-1}{x^{2}+x+4}).
a) Ta có ( y’=frac{(x^{2}+x+2)’.(x-1)-(x^{2}+x+2).(x-1)’}{(x-1)^{2}}) = ( frac{x^{2}-2x-3}{(x-1)^{2}})
Do đó, (y'<0Leftrightarrow frac{x^{2}-2x-3}{(x-1)^{2}})
( Leftrightarrow left{ matrix{ x ne 1 hfill cr – 1 < x < 3 hfill cr} right. Leftrightarrow )(x∈ (-1;1) ∪ (1;3)).
b) Ta có ( y’=frac{(x^{2}+3)’.(x+1)-(x^{2}+3).(x+1)’}{(x+1)^{2}}) = ( frac{x^{2}+2x-3}{(x+1)^{2}}).
Do đó, (y’≥0 Leftrightarrow frac{x^{2}+2x-3}{(x+1)^{2}}≥0 )
( Leftrightarrow left{ matrix{ x ne – 1 hfill cr left[ matrix{ x ge 1 hfill cr x le – 3 hfill cr} right. hfill cr} right. Leftrightarrow left[ matrix{ x ge 1 hfill cr x le – 3 hfill cr} right. Leftrightarrow x∈ (-∞;-3] ∪ [1;+∞)).
c).Ta có ( y’=frac{(2x-1)’.(x^{2}+x+4)-(2x-1).(x^{2}+x+4)’}{(x^{2}+x+4)}=frac{-2x^{2}+2x+9}{(x^{2}+x+4)}).
Do đó, (y’>0 Leftrightarrow frac{-2x^{2}+2x+9}{(x^{2}+x+4)} >0Leftrightarrow -2x^2+2x +9>0 )(Leftrightarrow frac{1-sqrt{19}}{2} < x < frac{1+sqrt{19}}{2}Leftrightarrow x∈ left ( frac{1-sqrt{19}}{2};frac{1+sqrt{19}}{2} right ))
Vì (x^2+x +4 =) ( left ( x+frac{1}{2} right )^{2})+ ( frac{15}{4} >0), với (∀ x ∈ mathbb R).
Bài 3: Tìm đạo hàm của các hàm số sau:
a) (y = 5sinx -3cosx);
b) ( y=frac{sinx+cosx}{sinx-cosx});
c) (y = x cotx);
d) (y = frac{sinx}{x}) + ( frac{x}{sinx});
e) (y = sqrt{(1 +2tan x)});
f) (y = sinsqrt{(1 +x^2)}).
a) (y’=5cosx-3(-sinx)=5cosx+3sinx);
b) ( y’={{(sinx+cos x)’.(sin x- cos x)-(sin x+cos x)(sin x-cos x)’}over{(sin x-cos x)^{2}}}) = ( {{(cos x-sin x)(sin x -cos x)-(sin x+ cos x)(cosx+sinx)}over{(sin x-cosx )^{2}}}) = ( {{-2}over{(sin x-cos x)^{2}}}).
c) (y’ = cotx +x. left ( -frac{1}{sin^{2}x} right )= cotx – frac{x}{sin^{2}x}).
d) ( y’=frac{(sin x)’.x-sin x.(x)’}{x^{2}}) +( frac{(x)’.sin x-x(sin x)’}{sin^{2}x}) = ( frac{x.cosx-sinx}{x^{2}}+frac{sin x-x.cosx}{sin^{2}x})( = (x. cosx -sinx) left ( frac{1}{x^{2}}-frac{1}{sin^{2}x} right )).
e) ( y’=frac{(1+2tanx)’}{2sqrt{1+2tanx}}) = ( frac{frac{2}{cos^{2}x}}{2sqrt{1+2tanx}}) = ( frac{1}{cos^{2}xsqrt{1+2tanx}}).
f) (y’ = (sqrt{(1+x^2)})’ cossqrt{(1+x^2)} )(= frac{(1+x^{2})’}{2sqrt{1+x^{2}}}cossqrt{(1+x^2)} = frac{x}{sqrt{1+x^{2}}}cossqrt{(1+x^2)}).
Bài 4: Tìm đạo hàm của các hàm số sau:
a) (y = left( {9 – 2x} right)(2{x^3} – 9{x^2} + 1));
b) (y = left ( 6sqrt{x} -frac{1}{x^{2}}right )(7x -3));
c) (y = (x -2)sqrt{(x^2+1)});
d) (y = tan^2x +cotx^2);
e) (y = cosfrac{x}{1+x}).
a) (y’ = left( {9 – 2x} right)'(2{x^3} – 9{x^2} + 1) + left( {9 – 2x} right)(2{x^3} – 9{x^2} + 1)’)
(= – 2(2{x^3} – 9{x^2} + 1) + left( {9 – 2x} right)(6{x^2} – 18x) )
(= – 16{x^3} + 108{x^2} – 162x – 2).
b) (y’ = left ( 6sqrt{x} -frac{1}{x^{2}}right )’.(7x -3) +left ( 6sqrt{x} -frac{1}{x^{2}}right )(7x -3)’)
(= left ( frac{3}{sqrt{x}} +frac{2}{x^{3}}right )(7x -3) +7 left ( 6sqrt{x} -frac{1}{x^{2}}right )).
c) (y’ = (x -2)’sqrt{(x^2+1)} + (x -2)sqrt {(x^2+1)}’ )
(= sqrt {(x^2+1)} + (x -2)frac{left ( x^{2}+1 right )’}{2sqrt{x^{2}+1}})
(= sqrt {(x^2+1)} + (x -2) frac{2x}{2sqrt{x^{2}+1}})
( = sqrt {(x^2+1)} + frac{x^{2}-2x}{sqrt{x^{2}+1}}) = ( frac{2x^{2}-2x+1}{sqrt{x^{2}+1}}).
d) (y’ = 2tanx.(tanx)’ – (x^2)’ left ( -frac{1}{sin^{2}x^{2}} right )) = ( frac{2tanx}{cos^{2}x}+frac{2x}{sin^{2}x^{2}}).
e) (y’ = left ( frac{1}{1+x} right )’sin frac{x}{1+x}) = ( -frac{1}{(1+x)^{2}}sin frac{x}{1+x}).
