Differentiability
DifferentiabilityA function is differentiable at a point if the curve is smooth continuous
i.e. lim limx a x a
f x f x
DifferentiabilityA function is differentiable at a point if the curve is smooth continuous
i.e. lim limx a x a
f x f x
y
x1
1
1y x
DifferentiabilityA function is differentiable at a point if the curve is smooth continuous
i.e. lim limx a x a
f x f x
y
x1
1
1y x
not differentiable at x = 1
DifferentiabilityA function is differentiable at a point if the curve is smooth continuous
i.e. lim limx a x a
f x f x
y
x1
1
1y x
1
lim 1x
f x
not differentiable at x = 1
DifferentiabilityA function is differentiable at a point if the curve is smooth continuous
i.e. lim limx a x a
f x f x
y
x1
1
1y x
1
lim 1x
f x
1
lim 1x
f x
not differentiable at x = 1
DifferentiabilityA function is differentiable at a point if the curve is smooth continuous
i.e. lim limx a x a
f x f x
y
x1
1
1y x
1
lim 1x
f x
1
lim 1x
f x
not differentiable at x = 1
y
x
2y x
DifferentiabilityA function is differentiable at a point if the curve is smooth continuous
i.e. lim limx a x a
f x f x
y
x1
1
1y x
1
lim 1x
f x
1
lim 1x
f x
not differentiable at x = 1
y
x
2y x
1
DifferentiabilityA function is differentiable at a point if the curve is smooth continuous
i.e. lim limx a x a
f x f x
y
x1
1
1y x
1
lim 1x
f x
1
lim 1x
f x
not differentiable at x = 1
y
x
2y x
1
differentiable at x = 1
Implicit Differentiation
Implicit Differentiationdf df dydx dy dx
Implicit Differentiationdf df dydx dy dx
2e.g. (i) x y
Implicit Differentiationdf df dydx dy dx
2e.g. (i) x y
2d dx y
dx dx
Implicit Differentiationdf df dydx dy dx
2e.g. (i) x y
2d dx y
dx dx
2 2d d dyy y
dx dy dx
Implicit Differentiationdf df dydx dy dx
2e.g. (i) x y
1 2dyydx
2d dx y
dx dx
2 2d d dyy y
dx dy dx
Implicit Differentiationdf df dydx dy dx
2e.g. (i) x y
1 2dyydx
2d dx y
dx dx
12
dydx y
2 2d d dyy y
dx dy dx
Implicit Differentiationdf df dydx dy dx
2e.g. (i) x y
1 2dyydx
2d dx y
dx dx
12
dydx y
2 2d d dyy y
dx dy dx
2 3(ii) dx y
dx
Implicit Differentiationdf df dydx dy dx
2e.g. (i) x y
1 2dyydx
2d dx y
dx dx
12
dydx y
2 2d d dyy y
dx dy dx
2 3(ii) dx y
dx
2 2 33 2dy
x y y xdx
Implicit Differentiationdf df dydx dy dx
2e.g. (i) x y
1 2dyydx
2d dx y
dx dx
12
dydx y
2 2d d dyy y
dx dy dx
2 3(ii) dx y
dx
2 2 33 2dy
x y y xdx
2 2 33 2dy
x y xydx
2 2 Find the equation of the tangent to 9 at the point 1,2 2iii x y
2 2 Find the equation of the tangent to 9 at the point 1,2 2iii x y 2 2 9x y
2 2 Find the equation of the tangent to 9 at the point 1,2 2iii x y 2 2 9x y
2 2 0dy
x ydx
2 2 Find the equation of the tangent to 9 at the point 1,2 2iii x y 2 2 9x y
2 2 0dy
x ydx
2 2dyy xdx
2 2 Find the equation of the tangent to 9 at the point 1,2 2iii x y 2 2 9x y
2 2 0dy
x ydx
2 2dyy xdx
dy xdx y
2 2 Find the equation of the tangent to 9 at the point 1,2 2iii x y 2 2 9x y
2 2 0dy
x ydx
2 2dyy xdx
dy xdx y
1at 1,2 2 ,
2 2
dydx
2 2 Find the equation of the tangent to 9 at the point 1,2 2iii x y 2 2 9x y
2 2 0dy
x ydx
2 2dyy xdx
dy xdx y
1at 1,2 2 ,
2 2
dydx
1 required slope
2 2
2 2 Find the equation of the tangent to 9 at the point 1,2 2iii x y 2 2 9x y
2 2 0dy
x ydx
2 2dyy xdx
dy xdx y
1at 1,2 2 ,
2 2
dydx
1 required slope
2 2
12 2 1
2 2y x
2 2 Find the equation of the tangent to 9 at the point 1,2 2iii x y 2 2 9x y
2 2 0dy
x ydx
2 2dyy xdx
dy xdx y
1at 1,2 2 ,
2 2
dydx
1 required slope
2 2
12 2 1
2 2y x
2 2 8 1y x
2 2 Find the equation of the tangent to 9 at the point 1,2 2iii x y 2 2 9x y
2 2 0dy
x ydx
2 2dyy xdx
dy xdx y
1at 1,2 2 ,
2 2
dydx
1 required slope
2 2
12 2 1
2 2y x
2 2 8 1y x
2 2 9 0x y
2 2 Find the equation of the tangent to 9 at the point 1,2 2iii x y 2 2 9x y
2 2 0dy
x ydx
2 2dyy xdx
dy xdx y
1at 1,2 2 ,
2 2
dydx
1 required slope
2 2
12 2 1
2 2y x
2 2 8 1y x
2 2 9 0x y
Exercise 7K; 1acegi, 2bdfh, 3a,4a, 7, 8
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