1. The Second Derivative

2. The Second Derivative
d2 d2y
y, f  x , 2  f  x , 2
dx dx

3. The Second Derivative d2 d2y
y, f  x , 2  f  x , 2
dx dx
e.g. i  y  3x 2  x  2

4. The Second Derivative d2 d2y
y, f  x , 2  f  x , 2
dx dx
e.g. i  y  3x 2  x  2
dy
 6x 1
dx

5. The Second Derivative d2 d2y
y, f  x , 2  f  x , 2
dx dx
e.g. i  y  3x 2  x  2
dy
 6x 1
dx
d2y
2
6
dx

6. The Second Derivative d2 d2y
y, f  x , 2  f  x , 2
dx dx
e.g. i  y  3x 2  x  2 ii  y  x 2  3
dy
 6x 1
dx
d2y
2
6
dx

7. The Second Derivative d2 d2y
y, f  x , 2  f  x , 2
dx dx
e.g. i  y  3x 2  x  2 ii  y  x 2  3
 x 2  3
1
dy
 6x 1 2
dx
d2y
2
6
dx

8. The Second Derivative d2 d2y
y, f  x , 2  f  x , 2
dx dx
e.g. i  y  3x 2  x  2 ii  y  x 2  3
 x 2  3
1
dy
 6x 1 2
dx
 x  3 2 2 x 
1
dy 1 2 
d2y
2
6 dx 2
dx

9. The Second Derivative d2 d2y
y, f  x , 2  f  x , 2
dx dx
e.g. i  y  3x 2  x  2 ii  y  x 2  3
 x 2  3
1
dy
 6x 1 2
dx
 x  3 2 2 x 
1
dy 1 2 
d2y
2
6 dx 2
 xx 2  3 2
dx 
1

10. The Second Derivative d2 d2y
y, f  x , 2  f  x , 2
dx dx
e.g. i  y  3x 2  x  2 ii  y  x 2  3
 x 2  3
1
dy
 6x 1 2
dx
 x  3 2 2 x 
1
dy 1 2 
d2y
2
6 dx 2
 xx 2  3 2
dx 
1
 1 2 
  x  x  3 2 2 x   x 2  3 2 1
d2y 
3

1
dx 2  2 

11. The Second Derivative d2 d2y
y, f  x , 2  f  x , 2
dx dx
e.g. i  y  3x 2  x  2 ii  y  x 2  3
 x 2  3
1
dy
 6x 1 2
dx
 x  3 2 2 x 
1
dy 1 2 
d2y
2
6 dx 2
 xx 2  3 2
dx 
1
 1 2 
  x  x  3 2 2 x   x 2  3 2 1
d2y 
3

1
dx 2  2 
  x 2 x 2  3  x 2  3
3 1
 
2 2

12. The Second Derivative d2 d2y
y, f  x , 2  f  x , 2
dx dx
e.g. i  y  3x 2  x  2 ii  y  x 2  3
 x 2  3
1
dy
 6x 1 2
dx
 x  3 2 2 x 
1
dy 1 2 
d2y
2
6 dx 2
 xx 2  3 2
dx 
1
 1 2 
  x  x  3 2 2 x   x 2  3 2 1
d2y 
3

1
dx 2  2 
  x 2 x 2  3  x 2  3
3 1
 
2 2
 x 2  3  x 2  x 2  3
3

2

13. The Second Derivative d2 d2y
y, f  x , 2  f  x , 2
dx dx
e.g. i  y  3x 2  x  2 ii  y  x 2  3
 x 2  3
1
dy
 6x 1 2
dx
 x  3 2 2 x 
1
dy 1 2 
d2y
2
6 dx 2
 xx 2  3 2
dx 
1
 1 2 
  x  x  3 2 2 x   x 2  3 2 1
d2y 
3

1
dx 2  2 
  x 2 x 2  3  x 2  3
3 1
 
2 2
 x 2  3  x 2  x 2  3
3

2
3
 2
x  3 x 2  3

14. Exercise 10D; 1 to 13, every 2nd in all