2. Differentiability
A function is differentiable at a point if the curve is smooth continuous
i.e. lim f x lim f x
x a x a
3. Differentiability
A function is differentiable at a point if the curve is smooth continuous
i.e. lim f x lim f x
x a x a
y
y x 1
1
1 x
4. Differentiability
A function is differentiable at a point if the curve is smooth continuous
i.e. lim f x lim f x
x a x a
y
y x 1
1
1 x
not differentiable at x = 1
5. Differentiability
A function is differentiable at a point if the curve is smooth continuous
i.e. lim f x lim f x
x a x a
y
y x 1
1
1 x
not differentiable at x = 1
lim f x 1
x1
6. Differentiability
A function is differentiable at a point if the curve is smooth continuous
i.e. lim f x lim f x
x a x a
y
y x 1
1
1 x
not differentiable at x = 1
lim f x 1
lim f x 1
x1 x1
7. Differentiability
A function is differentiable at a point if the curve is smooth continuous
i.e. lim f x lim f x
x a x a
y y y x2
y x 1
1
1 x x
not differentiable at x = 1
lim f x 1
lim f x 1
x1 x1
8. Differentiability
A function is differentiable at a point if the curve is smooth continuous
i.e. lim f x lim f x
x a x a
y y y x2
y x 1
1
1 x 1 x
not differentiable at x = 1
lim f x 1
lim f x 1
x1 x1
9. Differentiability
A function is differentiable at a point if the curve is smooth continuous
i.e. lim f x lim f x
x a x a
y y y x2
y x 1
1
1 x 1 x
not differentiable at x = 1 differentiable at x = 1
lim f x 1
lim f x 1
x1 x1
14. Implicit Differentiation
df df dy
dx dy dx
e.g. (i) x y 2
x y
d d 2
dx dx
d 2
dx
y dy y dx
d 2 dy
15. Implicit Differentiation
df df dy
dx dy dx
e.g. (i) x y 2
x y
d d 2
dx dx
dy
1 2y
dx
d 2
dx
y dy y dx
d 2 dy
16. Implicit Differentiation
df df dy
dx dy dx
e.g. (i) x y 2
x y
d d 2
dx dx
dy
1 2y
dx
dy 1
dx 2 y
d 2
dx
y dy y dx
d 2 dy
17. Implicit Differentiation
df df dy
dx dy dx
e.g. (i) x y 2 (ii)
dx
x y
d 2 3
x y
d d 2
dx dx
dy
1 2y
dx
dy 1
dx 2 y
d 2
dx
y dy y dx
d 2 dy
18. Implicit Differentiation
df df dy
dx dy dx
e.g. (i) x y 2 (ii)
dx
x y
d 2 3
x y
d d 2
x2 3 y 2 y3 2 x
dx dx dy
dy dx
1 2y
dx
dy 1
dx 2 y
d 2
dx
y dy y dx
d 2 dy
19. Implicit Differentiation
df df dy
dx dy dx
e.g. (i) x y 2 (ii)
dx
x y
d 2 3
x y
d d 2
x2 3 y 2 y3 2 x
dx dx dy
dy dx
1 2y 2 2 dy
dx 3x y 2 xy 3
dy 1 dx
dx 2 y
d 2
dx
y dy y dx
d 2 dy
20. iii Find the equation of the tangent to x 2 y 2 9 at the point 1, 2 2
21. iii Find the equation of the tangent to x 2 y 2 9 at the point 1, 2 2
x2 y 2 9
22. iii Find the equation of the tangent to x 2 y 2 9 at the point 1, 2 2
x2 y 2 9
dy
2x 2 y 0
dx
23. iii Find the equation of the tangent to x 2 y 2 9 at the point 1, 2 2
x2 y 2 9
dy
2x 2 y 0
dx
dy
2 y 2 x
dx
24. iii Find the equation of the tangent to x 2 y 2 9 at the point 1, 2 2
x2 y 2 9
dy
2x 2 y 0
dx
dy
2 y 2 x
dx
dy x
dx y
25. iii Find the equation of the tangent to x 2 y 2 9 at the point 1, 2 2
x2 y 2 9
dy
2x 2 y 0
dx
dy
2 y 2 x
dx
dy x
dx y
dy
at 1, 2 2 ,
dx
1
2 2
26. iii Find the equation of the tangent to x 2 y 2 9 at the point 1, 2 2
x2 y 2 9
dy
2x 2 y 0
dx
dy
2 y 2 x
dx
dy x
dx y
dy
at 1, 2 2 ,
dx
1
2 2
1
required slope
2 2
27. iii Find the equation of the tangent to x 2 y 2 9 at the point 1, 2 2
x2 y 2 9 1
dy y2 2 x 1
2x 2 y 0 2 2
dx
dy
2 y 2 x
dx
dy x
dx y
dy
at 1, 2 2 ,
dx
1
2 2
1
required slope
2 2
28. iii Find the equation of the tangent to x 2 y 2 9 at the point 1, 2 2
x2 y 2 9 1
dy y2 2 x 1
2x 2 y 0 2 2
dx
2 2y 8 x 1
dy
2 y 2 x
dx
dy x
dx y
dy
at 1, 2 2 ,
dx
1
2 2
1
required slope
2 2
29. iii Find the equation of the tangent to x 2 y 2 9 at the point 1, 2 2
x2 y 2 9 1
dy y2 2 x 1
2x 2 y 0 2 2
dx
2 2y 8 x 1
dy
2 y 2 x
dx x 2 2y 9 0
dy x
dx y
dy
at 1, 2 2 ,
dx
1
2 2
1
required slope
2 2
30. iii Find the equation of the tangent to x 2 y 2 9 at the point 1, 2 2
x2 y 2 9 1
dy y2 2 x 1
2x 2 y 0 2 2
dx
2 2y 8 x 1
dy
2 y 2 x
dx x 2 2y 9 0
dy x
dx y
dy
at 1, 2 2 ,
dx
1
2 2
1 Exercise 7K; 1acegi, 2bdfh, 3a,
required slope
2 2 4a, 7, 8