Đạo hàm và phương trình y'=0

[r]

1/> y:= 2*sin(x)+cos(2*x);

:=

y 2sin x( )  cos 2 x( )

> y1:=diff(y,x);

:=

y1 2cos x( )  2sin 2 x( )

> y2:=solve(y1=0,{x});

:=

2 {x  1 }

2 {x  1 }

6 {x  5 }

6 2/> y:= (1/3*(sin(x)^3)-(cos(x))^2);

:=

3sin x( )

3 cos x( )2

> y1:=diff(y,x);

:=

y1 sin x( )2cos x( )  2cos x( )sin x( )

> y2:=solve(y1=0,{x});

{x 0  } {x  1 },

2 {x  1 }

2 3/ > y:=

(1/3*(cos(x)^3)+(sin(x))^2);

:=

3cos x( )3 sin x( )2

> y1:=diff(y,x);

:=

y1 cos x( )2sin x( )  2cos x( )sin x( )

> y2:=solve(y1=0,{x});

{x 0  } {x  1 }

2 {x  1 }

2 4/ > y:=

(1/3*(cos(x)^4)+(sin(x))^4);

:=

3cos x( )

4 sin x( )4

> y1:=diff(y,x);

:=

3cos x( )3sin x( ) 4sin x( )3cos x( )

> y2:=solve(y1=0,{x});

{x 0  } {x  1 }

2 {x  1 }

2 {x  1 },

6 {x  1 }

6

5/> y:= (1/3*(cos(x)^4)-(sin(x))^4);

3cos x( )

4 sin x( )4

> y1:=diff(y,x);

:=

3cos x( )3sin x( ) 4sin x( )3cos x( )

> y2:=solve(y1=0,{x});

{x 0  } {x  1 }

2 {x  1 }

2

6/ > y:= (cos(x)^4)-(sin(x))^4;

:=

y cos x( )4  sin x( )4

> y1:=diff(y,x);

:=

y1 4cos x( )3sin x( )  4sin x( )3cos x( )

> y2:=solve(y1=0,{x});

y2 := {x 0  },{x  1 },

2 {x  1 }

2

7/ > y:= (cos(x)^6)+(sin(x))^6;

:=

y cos x( )6  sin x( )6

> y1:=diff(y,x);

:=

y1 6cos x( )5sin x( )  6sin x( )5cos x( )

> y2:=solve(y1=0,{x});

{x 0  } {x  1 }

2 {x  1 }

2 ,

{x  1 }

4 {x  1 }

4

8/ > y:= 3*sin(x)+sin(3*x);

:=

y 3sin x( )  sin 3 x( )

> y1:=diff(y,x);

:=

y1 3cos x( )  3cos 3 x( )

> y2:=solve(y1=0,{x});

:=

2 {x  1 }

4 {x  3 }

4

9/ > y:= 3*sin(x)-sin(3*x);

:=

y 3sin x( )  sin 3 x( )

> y1:=diff(y,x);

:=

y1 3cos x( )  3cos 3 x( )

> y2:=solve(y1=0,{x});

:=

2 {x 0  } {x  }

10/ > y:= sin(2*x)/(2+cos(2*x));

:=

y 2  sin 2 x cos 2 x( ( ) )

> y1:=diff(y,x);

:=

y1 2 cos 2 x( ) 



2 cos 2 x( )

2sin 2 x( )2 (2  cos 2 x( ))2

> y2:=simplify(y1);

:=

y2 2 2cos 2 x( )  1

4 4cos 2 x( ) cos 2 x( )2

> y3:=solve(y2=0,{x});

:=

3

11/ > y:= cos(2*x)/(2+sin(2*x));

:=

y 2  cos 2 x sin 2 x( ( ) )

> y1:=diff(y,x);

:=

y1 2 2  sin 2 x sin 2 x( ( ) ) 

2cos 2 x( )2

(2  sin 2 x( ))2

> y2:=simplify(y1);

:=

y2 2 2sin 2 x( )  1

  5 4sin 2 x( )  cos 2 x( )2

> y3:=solve(y2=0,{x});

:=

y3 {x   1 },

12 {x   5 }

12

12/ > y:=sqrt(

cos(1*x))+sqrt(sin(x));

:=

y cos x( )  sin x( )

> y1:=diff(y,x);

:=

2

( )

sin x

( )

cos x

1

2cos x( ) ( )

sin x

> y2:=simplify(y1);

:=

2

sin x( )(3 2/ )  cos x( )(3 2/ )

( )

cos x sin x( )

> y3:=solve(y2=0,{x});

:=

y3 {x  1 },

4 {x  3 }

4

13/ > y:=(sin(x))^2 + 3*cos(2*x);

:=

y sin x( )2  3cos 2 x( )

> y1:=diff(y,x);

:=

y1 2sin x( )cos x( )  6sin 2 x( )

> y2:=simplify(y1);

:=

y2 10sin x( )cos x( )

> y3:=solve(y2=0,{x});

:=

y3 {x  1 },

2 {x 0  }

14/ > y:=e^(x)*sin(x);

:=

y e x sin x( )

> y1:=diff(y,x);

:=

y1 e x ln e( )sin x( )  e x cos x( )

> y2:=simplify(y1);

:=

y2 e x (ln e( )sin x( )  cos x( ))

> y3:=solve(y2=0,{x});

:=



 

arctan 1

( )

ln e

15/ > y:=e^(x)*cos(x);

:=

y e x cos x( )

> y1:=diff(y,x);

:=

y1 e x ln e( )cos x( )  e x sin x( )

> y2:=simplify(y1);

:=

y2 e x (ln e( )cos x( )  sin x( ))

> y3:=solve(y2=0,{x});

:=

y3 {x  arctan(ln e( ))}

16/ > y:=e^(x)*(x-1);

:=

y e x(x 1  )

> y1:=diff(y,x);

:=

y1 e x ln e (( ) x 1  )  e x

> y2:=simplify(y1);

:=

y2 e x ln e x e( )  x ln e( )  e x

> y3:=solve(y2=0,{x});

:=

y3 {x  ln e( )  1}

( )

ln e

17/ > y:=(x)*e^(1-x);

:=

y x e(1 x  )

> y1:=diff(y,x);

:=

y1 e(1 x  )  x e(1 x  )ln e( )

> y2:=simplify(y1);

:=

y2 e(1 x  )  x e(1 x  )ln e( )

> y3:=solve(y2=0,{x});

:=

( )

ln e

18/ > y:=(x)*e^(x-1);

:=

y x e(x  1)

> y1:=diff(y,x);

:=

y1 e(x  1)  x e(x  1)ln e( )

> y2:=simplify(y1);

:=

y2 e(x  1)  x e(x  1)ln e( )

> y3:=solve(y2=0,{x});

:=

( )

ln e

19/ > y:=(x)^2*e^(x^2);

:=

y x2e(x2)

> y1:=diff(y,x);

:=

y1 2 x e(x2)  2 x3e(x2)ln e( )

> y2:=simplify(y1);

:=

y2 2 x e(x2)  2 x3e(x2)ln e( )

> y3:=solve(y2=0,{x});

:=

y3 {x 0  },{x   1 },

 ( )ln e {x  1 }

 ( )ln e

20/ > y:=(x)^2*e^(x^2-1);

:=

y x2e(x2 1  )

> y1:=diff(y,x);

:=

y1 2 x e(x2 1  )  2 x3e(x2 1  )ln e( )

> y2:=solve(y1=0,{x});

:=

y2 {x 0  },{x   1 },

 ( )ln e {x  1 }

 ( )ln e

21/ > y:=(x)^2*e^(x);

:=

y x2e x

> y1:=diff(y,x);

:=

y1 2 x e x  x2e x ln e( )

> y2:=solve(y1=0,{x});

:=

y2 {x 0  },{x  2 1 }

( )

ln e

22/ > y:=(x)^2*e^(x-1);

:=

y x2e(x  1)

> y1:=diff(y,x);

:=

y1 2 x e(x  1)  x2e(x  1)ln e( )

> y2:=solve(y1=0,{x});

:=

y2 {x 0  },{x  2 1 }

( )

ln e

23/ > y:=(x)*e^(sqrt(x));

:=

y x e( x)

> y1:=diff(y,x);

:=

y1 e( x)  1

2 x e

( x)

( )

ln e

> y2:=solve(y1=0,{x});

:=

( )

ln e 2

24/ > y:=(sqrt(4+x))+(sqrt(4-x));

:=

y 4 x   4 x 

> y1:=diff(y,x);

:=

2

1 

4 x

1 2

1 

4 x

> y2:=solve(y1=0,{x});

:=

y2 {x 0  }

25 / > y:=(sqrt(4+x^2))-(sqrt(x^2));

:=

y 4 x  2  x2

> y1:=diff(y,x);

:=



4 x2

x

x2

> y2:=solve(y1 =0,{x});

x = 0

26/ > y:=(sqrt(4-x^2))+(sqrt(x^2));

:=

y 4 x  2  x2

> y1:=diff(y,x);

:=



4 x2

x

x2

> y2:=solve(y1 =0,{x});

:=

y2 {x  2},{x   2}

x = 0

27/ > restart:

> y:= x+(sqrt(2-x^2));

:=

y x  2 x  2

> y1:=diff(y,x);

:=



2 x2

> y2:=solve(y1 =0,{x});

:=

y2 {x 1  }

28/ > y:= 3*x-5*(sqrt(4+x^2));

:=

y 3 x 5  4 x  2

> y1:=diff(y,x);

:=



4 x2

> y2:=solve(y1 =0,{x});

:=

y2 {x  3}

2 29/> y:= 6*x-8*(sqrt(4*x-x^2));

:=

y 6 x 8  4 x x  2

> y1:=diff(y,x);

:=

y1 6  4 (4 2 x  )



4 x x2

> y2:=solve(y1 =0,{x});

:=

y2 {x  4}

5

30/ > y:= 3*sqrt(4-x)+6*sqrt(x+6);

:=

y 3 4 x   6 x 6 

> y1:=diff(y,x);

:=

2

1 

4 x

3 

x 6

> y2:=solve(y1 =0,{x});

:=

y2 {x 2  }

31/ > y:= 3*sqrt(9-x)+6*sqrt(x+6);

:=

y 3 9 x   6 x 6 

> y1:=diff(y,x);

:=

2

1 

9 x

3 

x 6

> y2:=solve(y1 =0,{x});

:=

y2 {x 6  }

32/ > y:= 3*sqrt(12-x)+6*sqrt(x+8);

:=

y 3 12 x   6 x 8 

> y1:=diff(y,x);

:=

2

1 

12 x

3 

x 8

> y2:=solve(y1 =0,{x});

:=

y2 {x 8  }

33/ > y:= x*ln(x);

:=

y x ( ) ln x

> y1:=diff(y,x);

:=

y1 ln x( )  1

> y2:=solve(y1 =0,{x});

:=

y2 {x e  (-1)} 34/ > y:= ln(x)/x;

:=

y ln x( ) x

> y1:=diff(y,x);

:=

x2

( )

ln x

x2

> y2:=solve(y1 =0,{x});

:=

y2 {x e  } 35/ > y:= ln(x)/x^2;

:=

y ln x( )

x2

> y1:=diff(y,x);

:=

x3

2 ( )ln x

x3

> y2:=solve(y1 =0,{x});

:=

y2 {x e  (1 2/ )}

36/ > y:= ln(x)*x^2;

:=

y ln x x( ) 2

> y1:=diff(y,x);

:=

y1 x 2 x ( )  ln x

> y2:=solve(y1 =0,{x});

:=

y2 {x e  (-1 2/ )}

37/ > y:=( x^2)*ln(x^2);

:=

y x2ln x( )2

> y1:=diff(y,x);

:=

y1 2 x ( ) ln x2  2 x

> y2:=solve(y1 =0,{x});

:=

y2 {x   e(-1)},{x  e(-1)}

38/ > y:=1*x-ln(x^2-1*x+1);

:=

y x  ln(x2   x 1)

> y1:=diff(y,x);

:=

y1 1  2 x 1 

 

> y2:=solve(y1=0,{x});

:=

y2 {x 2  },{x 1  }

39/ > y:=1*x-2*ln(5*x^2+5*x+10);

:=

y x 2 (  ln 5 x2  5 x 10  )

> y1:=diff(y,x);

:=

y1 1  2 (10 x 5  )

 

5 x2 5 x 10

> y2:=solve(y1=0,{x});

:=

y2 {x 0  },{x 3  }

40/ > y:=1*x+2*ln(5*x^2+5*x+10);

:=

y x 2 (  ln 5 x2  5 x 10  )

> y1:=diff(y,x);

:=

y1 1  2 (10 x 5  )

 

5 x2 5 x 10

> y2:=solve(y1=0,{x});

:=

y2 {x -1  },{x -4  }

41/ > y:=1*x-1*ln(5*x^2-10*x+10);

:=

y x  ln(5 x2  10 x 10  )

> y1:=diff(y,x);

:=

y1 1  10 x 10 

 

5 x2 10 x 10

> y2:=solve(y1=0,{x});

:=

y2 {x 2  },{x 2  }

42/ > y:=1*x-1*ln(1*x^2-10*x+10);

:=

y x  ln(x2  10 x 10  )

> y1:=diff(y,x);

:=

y1 1  2 x 10 

 

x2 10 x 10

> y2:=solve(y1=0,{x});

:=

y2 {x 10  },{x 2  }

43/ > y:=1*x*e^(-x^2/2);

:=

y x e( /1 2 x2)

> y1:=diff(y,x);

:=

y1 e( /1 2 x2)  x2e( /1 2 x2)ln e( )

> y2:=solve(y1=0,{x});

:=

( )

ln e {x   1 }

( )

ln e

44/ > y:=tan(x)+cot(x);

:=

y tan x( )  cot x( )

> y1:=diff(y,x);

:=

y1 tan x( )2  cot x( )2

> y2:=solve(y1=0,{x});

:=

y2 {x  1 },

4 {x  1 }

4