1. Differentiability

2. Differentiability
A function is differentiable at a point if the curve is smooth continuous
   i.e. lim lim
x a x a
f x f x 
 
 

3. Differentiability
A function is differentiable at a point if the curve is smooth continuous
   i.e. lim lim
x a x a
f x f x 
 
 
y
x
1
1
1y x 

4. Differentiability
A function is differentiable at a point if the curve is smooth continuous
   i.e. lim lim
x a x a
f x f x 
 
 
y
x
1
1
1y x 
not differentiable at x = 1

5. Differentiability
A function is differentiable at a point if the curve is smooth continuous
   i.e. lim lim
x a x a
f x f x 
 
 
y
x
1
1
1y x 
 1
lim 1
x
f x

  
not differentiable at x = 1

6. Differentiability
A function is differentiable at a point if the curve is smooth continuous
   i.e. lim lim
x a x a
f x f x 
 
 
y
x
1
1
1y x 
 1
lim 1
x
f x

    1
lim 1
x
f x

 
not differentiable at x = 1

7. Differentiability
A function is differentiable at a point if the curve is smooth continuous
   i.e. lim lim
x a x a
f x f x 
 
 
y
x
1
1
1y x 
 1
lim 1
x
f x

    1
lim 1
x
f x

 
not differentiable at x = 1
y
x
2
y x

8. Differentiability
A function is differentiable at a point if the curve is smooth continuous
   i.e. lim lim
x a x a
f x f x 
 
 
y
x
1
1
1y x 
 1
lim 1
x
f x

    1
lim 1
x
f x

 
not differentiable at x = 1
y
x
2
y x
1

9. Differentiability
A function is differentiable at a point if the curve is smooth continuous
   i.e. lim lim
x a x a
f x f x 
 
 
y
x
1
1
1y x 
 1
lim 1
x
f x

    1
lim 1
x
f x

 
not differentiable at x = 1
y
x
2
y x
1
differentiable at x = 1

10. Implicit Differentiation

11. Implicit Differentiation
df df dy
dx dy dx
 

12. Implicit Differentiation
df df dy
dx dy dx
 
2
e.g. (i) x y

13. Implicit Differentiation
df df dy
dx dy dx
 
2
e.g. (i) x y
   2d d
x y
dx dx


14. Implicit Differentiation
df df dy
dx dy dx
 
2
e.g. (i) x y
   2d d
x y
dx dx

   2 2d d dy
y y
dx dy dx
 

15. Implicit Differentiation
df df dy
dx dy dx
 
2
e.g. (i) x y
1 2
dy
y
dx

   2d d
x y
dx dx

   2 2d d dy
y y
dx dy dx
 

16. Implicit Differentiation
df df dy
dx dy dx
 
2
e.g. (i) x y
1 2
dy
y
dx

   2d d
x y
dx dx

1
2
dy
dx y

   2 2d d dy
y y
dx dy dx
 

17. Implicit Differentiation
df df dy
dx dy dx
 
2
e.g. (i) x y
1 2
dy
y
dx

   2d d
x y
dx dx

1
2
dy
dx y

   2 2d d dy
y y
dx dy dx
 
 2 3
(ii)
d
x y
dx

18. Implicit Differentiation
df df dy
dx dy dx
 
2
e.g. (i) x y
1 2
dy
y
dx

   2d d
x y
dx dx

1
2
dy
dx y

   2 2d d dy
y y
dx dy dx
 
 2 3
(ii)
d
x y
dx
    2 2 3
3 2
dy
x y y x
dx
   
 

19. Implicit Differentiation
df df dy
dx dy dx
 
2
e.g. (i) x y
1 2
dy
y
dx

   2d d
x y
dx dx

1
2
dy
dx y

   2 2d d dy
y y
dx dy dx
 
 2 3
(ii)
d
x y
dx
    2 2 3
3 2
dy
x y y x
dx
   
 
2 2 3
3 2
dy
x y xy
dx
 

20.    2 2
Find the equation of the tangent to 9 at the point 1,2 2iii x y 

21.    2 2
Find the equation of the tangent to 9 at the point 1,2 2iii x y 
2 2
9x y 

22.    2 2
Find the equation of the tangent to 9 at the point 1,2 2iii x y 
2 2
9x y 
2 2 0
dy
x y
dx
 

23.    2 2
Find the equation of the tangent to 9 at the point 1,2 2iii x y 
2 2
9x y 
2 2 0
dy
x y
dx
 
2 2
dy
y x
dx
 

24.    2 2
Find the equation of the tangent to 9 at the point 1,2 2iii x y 
2 2
9x y 
2 2 0
dy
x y
dx
 
2 2
dy
y x
dx
 
dy x
dx y
 

25.    2 2
Find the equation of the tangent to 9 at the point 1,2 2iii x y 
2 2
9x y 
2 2 0
dy
x y
dx
 
2 2
dy
y x
dx
 
dy x
dx y
 
  1
at 1,2 2 ,
2 2
dy
dx
 

26.    2 2
Find the equation of the tangent to 9 at the point 1,2 2iii x y 
2 2
9x y 
2 2 0
dy
x y
dx
 
2 2
dy
y x
dx
 
dy x
dx y
 
  1
at 1,2 2 ,
2 2
dy
dx
 
1
required slope
2 2
  

27.    2 2
Find the equation of the tangent to 9 at the point 1,2 2iii x y 
2 2
9x y 
2 2 0
dy
x y
dx
 
2 2
dy
y x
dx
 
dy x
dx y
 
  1
at 1,2 2 ,
2 2
dy
dx
 
1
required slope
2 2
  
 
1
2 2 1
2 2
y x   

28.    2 2
Find the equation of the tangent to 9 at the point 1,2 2iii x y 
2 2
9x y 
2 2 0
dy
x y
dx
 
2 2
dy
y x
dx
 
dy x
dx y
 
  1
at 1,2 2 ,
2 2
dy
dx
 
1
required slope
2 2
  
 
1
2 2 1
2 2
y x   
2 2 8 1y x   

29.    2 2
Find the equation of the tangent to 9 at the point 1,2 2iii x y 
2 2
9x y 
2 2 0
dy
x y
dx
 
2 2
dy
y x
dx
 
dy x
dx y
 
  1
at 1,2 2 ,
2 2
dy
dx
 
1
required slope
2 2
  
 
1
2 2 1
2 2
y x   
2 2 8 1y x   
2 2 9 0x y  

30.    2 2
Find the equation of the tangent to 9 at the point 1,2 2iii x y 
2 2
9x y 
2 2 0
dy
x y
dx
 
2 2
dy
y x
dx
 
dy x
dx y
 
  1
at 1,2 2 ,
2 2
dy
dx
 
1
required slope
2 2
  
 
1
2 2 1
2 2
y x   
2 2 8 1y x   
2 2 9 0x y  
Exercise 7K; 1acegi, 2bdfh, 3a,
4a, 7, 8