The document discusses differentiability and implicit differentiation. It defines differentiability as a function having a smooth, continuous curve at a point. It provides examples of determining if functions are differentiable at various points. It then introduces implicit differentiation and uses it to find the derivative of implicitly defined functions. It works through examples of using implicit differentiation to find derivatives and the equation of a tangent line to an implicitly defined curve at a given point.

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