11X1 T09 08 implicit differentiation (2011)

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 ydx dx

Implicit Differentiationdf df dydx dy dx

2e.g. (i) x y

2d dx ydx dx

2 2d d dyy ydx dy dx

Implicit Differentiationdf df dydx dy dx

2e.g. (i) x y

1 2 dyydx

2d dx ydx dx

2 2d d dyy ydx dy dx

Implicit Differentiationdf df dydx dy dx

2e.g. (i) x y

1 2 dyydx

2d dx ydx dx

12

dydx y

2 2d d dyy ydx dy dx

Implicit Differentiationdf df dydx dy dx

2e.g. (i) x y

1 2 dyydx

2d dx ydx dx

12

dydx y

2 2d d dyy ydx dy dx

2 3(ii) d x ydx

Implicit Differentiationdf df dydx dy dx

2e.g. (i) x y

1 2 dyydx

2d dx ydx dx

12

dydx y

2 2d d dyy ydx dy dx

2 3(ii) d x ydx

2 2 33 2dyx y y xdx

Implicit Differentiationdf df dydx dy dx

2e.g. (i) x y

1 2 dyydx

2d dx ydx dx

12

dydx y

2 2d d dyy ydx dy dx

2 3(ii) d x ydx

2 2 33 2dyx y y xdx

2 2 33 2dyx 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 0dyx 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 0dyx 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 0dyx 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 0dyx 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 0dyx ydx

2 2dyy xdx

dy xdx y

1at 1,2 2 ,2 2

dydx

1 required slope2 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 0dyx ydx

2 2dyy xdx

dy xdx y

1at 1,2 2 ,2 2

dydx

1 required slope2 2

12 2 12 2

y 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 0dyx ydx

2 2dyy xdx

dy xdx y

1at 1,2 2 ,2 2

dydx

1 required slope2 2

12 2 12 2

y 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 0dyx ydx

2 2dyy xdx

dy xdx y

1at 1,2 2 ,2 2

dydx

1 required slope2 2

12 2 12 2

y 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 0dyx ydx

2 2dyy xdx

dy xdx y

1at 1,2 2 ,2 2

dydx

1 required slope2 2

12 2 12 2

y x

2 2 8 1y x

2 2 9 0x y

Exercise 7K; 1acegi, 2bdfh, 3a,4a, 7, 8