11x1 t08 03 rules for differentiation
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Rules For Differentiation
Rules For Differentiation(1) y c
Rules For Differentiation(1) y c
f x c
Rules For Differentiation(1) y c
f x c
f x h c
Rules For Differentiation(1) y c
f x c
f x h c
0
limh
c cf xh
Rules For Differentiation(1) y c
f x c
f x h c
0
limh
c cf xh
0
0limh h
Rules For Differentiation(1) y c
f x c
f x h c
0
limh
c cf xh
0
0limh h
0lim0
0h
Rules For Differentiation(1) y c
f x c
f x h c
0
limh
c cf xh
0
0limh h
0lim0
0h
(2) y kx
Rules For Differentiation(1) y c
f x c
f x h c
0
limh
c cf xh
0
0limh h
0lim0
0h
(2) y kx
f x kx
Rules For Differentiation(1) y c
f x c
f x h c
0
limh
c cf xh
0
0limh h
0lim0
0h
(2) y kx
f x kx
f x h k x hkx kh
Rules For Differentiation(1) y c
f x c
f x h c
0
limh
c cf xh
0
0limh h
0lim0
0h
(2) y kx
f x kx
f x h k x hkx kh
0
limh
kx kh kxf xh
Rules For Differentiation(1) y c
f x c
f x h c
0
limh
c cf xh
0
0limh h
0lim0
0h
(2) y kx
f x kx
f x h k x hkx kh
0
limh
kx kh kxf xh
0limh
khh
Rules For Differentiation(1) y c
f x c
f x h c
0
limh
c cf xh
0
0limh h
0lim0
0h
(2) y kx
f x kx
f x h k x hkx kh
0
limh
kx kh kxf xh
0limh
khh
0limh
k
k
(3) ny x
(3) ny x
2 2a b a b a b
(3) ny x
2 2a b a b a b
3 3 2 2a b a b a ab b
(3) ny x
2 2a b a b a b
3 3 2 2a b a b a ab b
4 4 3 2 2 3a b a b a a b ab b
(3) ny x
2 2a b a b a b
3 3 2 2a b a b a ab b
4 4 3 2 2 3a b a b a a b ab b
1 2 3 2 2 3 2 1n n n n n n n na b a b a a b a b a b ab b
(3) ny x
2 2a b a b a b
3 3 2 2a b a b a ab b
4 4 3 2 2 3a b a b a a b ab b
1 2 3 2 2 3 2 1n n n n n n n na b a b a a b a b a b ab b
nf x x
(3) ny x
2 2a b a b a b
3 3 2 2a b a b a ab b
4 4 3 2 2 3a b a b a a b ab b
1 2 3 2 2 3 2 1n n n n n n n na b a b a a b a b a b ab b
nf x x nf x h x h
(3) ny x
2 2a b a b a b
3 3 2 2a b a b a ab b
4 4 3 2 2 3a b a b a a b ab b
1 2 3 2 2 3 2 1n n n n n n n na b a b a a b a b a b ab b
nf x x nf x h x h
0
limn n
h
x h xf x
h
(3) ny x
2 2a b a b a b
3 3 2 2a b a b a ab b
4 4 3 2 2 3a b a b a a b ab b
1 2 3 2 2 3 2 1n n n n n n n na b a b a a b a b a b ab b
nf x x nf x h x h
0
limn n
h
x h xf x
h
1 2 2 1
0lim
n n n n
h
x h x x h x h x x h x x
h
(3) ny x
2 2a b a b a b
3 3 2 2a b a b a ab b
4 4 3 2 2 3a b a b a a b ab b
1 2 3 2 2 3 2 1n n n n n n n na b a b a a b a b a b ab b
nf x x nf x h x h
0
limn n
h
x h xf x
h
1 2 2 1
0lim
n n n n
h
x h x x h x h x x h x x
h
1 2 2 1
0lim
n n n n
h
h x h x h x x h x x
h
(3) ny x
2 2a b a b a b
3 3 2 2a b a b a ab b
4 4 3 2 2 3a b a b a a b ab b
1 2 3 2 2 3 2 1n n n n n n n na b a b a a b a b a b ab b
nf x x nf x h x h
0
limn n
h
x h xf x
h
1 2 2 1
0lim
n n n n
h
x h x x h x h x x h x x
h
1 2 2 1
0lim
n n n n
h
h x h x h x x h x x
h
1 2 2 1
0lim n n n n
hx h x h x x h x x
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x 1nnx
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x 1nnx
1(4) yx
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x 1nnx
1(4) yx
1f xx
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x 1nnx
1(4) yx
1f xx
1f x hx h
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x 1nnx
1(4) yx
1f xx
1f x hx h
0
1 1
limh
x h xf xh
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x 1nnx
1(4) yx
1f xx
1f x hx h
0
1 1
limh
x h xf xh
0limh
x x hhx x h
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x 1nnx
1(4) yx
1f xx
1f x hx h
0
1 1
limh
x h xf xh
0limh
x x hhx x h
0limh
hhx x h
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x 1nnx
1(4) yx
1f xx
1f x hx h
0
1 1
limh
x h xf xh
0limh
x x hhx x h
0limh
hhx x h
0
1limh x x h
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x 1nnx
1(4) yx
1f xx
1f x hx h
0
1 1
limh
x h xf xh
0limh
x x hhx x h
0limh
hhx x h
0
1limh x x h
21
x
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x 1nnx
1(4) yx
1f xx
1f x hx h
0
1 1
limh
x h xf xh
0limh
x x hhx x h
0limh
hhx x h
0
1limh x x h
21
x
Note:
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x 1nnx
1(4) yx
1f xx
1f x hx h
0
1 1
limh
x h xf xh
0limh
x x hhx x h
0limh
hhx x h
0
1limh x x h
21
x
Note:
1f x x
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x 1nnx
1(4) yx
1f xx
1f x hx h
0
1 1
limh
x h xf xh
0limh
x x hhx x h
0limh
hhx x h
0
1limh x x h
21
x
Note:
1f x x
2f x x
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x 1nnx
1(4) yx
1f xx
1f x hx h
0
1 1
limh
x h xf xh
0limh
x x hhx x h
0limh
hhx x h
0
1limh x x h
21
x
Note:
1f x x
2f x x
21
x
(5) y x
(5) y x
f x x
(5) y x
f x x
f x h x h
(5) y x
f x x
f x h x h 0
limh
x h xf xh
(5) y x
f x x
f x h x h 0
limh
x h xf xh
x h xx h x
(5) y x
f x x
f x h x h 0
limh
x h xf xh
x h xx h x
0limh
x h xh x h x
(5) y x
f x x
f x h x h 0
limh
x h xf xh
x h xx h x
0limh
x h xh x h x
0limh
hh x h x
(5) y x
f x x
f x h x h 0
limh
x h xf xh
x h xx h x
0limh
x h xh x h x
0limh
hh x h x
0
1limh x h x
(5) y x
f x x
f x h x h 0
limh
x h xf xh
x h xx h x
0limh
x h xh x h x
0limh
hh x h x
0
1limh x h x
1x x
(5) y x
f x x
f x h x h 0
limh
x h xf xh
x h xx h x
0limh
x h xh x h x
0limh
hh x h x
0
1limh x h x
1x x
12 x
(5) y x
f x x
f x h x h 0
limh
x h xf xh
x h xx h x
0limh
x h xh x h x
0limh
hh x h x
0
1limh x h x
1x x
12 x
Note:
(5) y x
f x x
f x h x h 0
limh
x h xf xh
x h xx h x
0limh
x h xh x h x
0limh
hh x h x
0
1limh x h x
1x x
12 x
Note:
12f x x
(5) y x
f x x
f x h x h 0
limh
x h xf xh
x h xx h x
0limh
x h xh x h x
0limh
hh x h x
0
1limh x h x
1x x
12 x
Note:
12f x x
121
2f x x
(5) y x
f x x
f x h x h 0
limh
x h xf xh
x h xx h x
0limh
x h xh x h x
0limh
hh x h x
0
1limh x h x
1x x
12 x
Note:
12f x x
121
2f x x
12 x
e.g. 7i y
e.g. 7i y
0dydx
e.g. 7i y
0dydx
37ii y x
e.g. 7i y
0dydx
37ii y x
37dydx
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
21 3vi y xx
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
21 3vi y xx
23x x
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
21 3vi y xx
23x x 33 2dy x
dx
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
21 3vi y xx
23x x 33 2dy x
dx
323x
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
21 3vi y xx
23x x 33 2dy x
dx
323x
2 vii y x x
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
21 3vi y xx
23x x 33 2dy x
dx
323x
2 vii y x x52x
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
21 3vi y xx
23x x 33 2dy x
dx
323x
2 vii y x x52x
325
2dy xdx
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
21 3vi y xx
23x x 33 2dy x
dx
323x
2 vii y x x52x
325
2dy xdx
52
x x
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
21 3vi y xx
23x x 33 2dy x
dx
323x
2 vii y x x52x
325
2dy xdx
52
x x
3If 3, find 2viii f x x
f
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
21 3vi y xx
23x x 33 2dy x
dx
323x
2 vii y x x52x
325
2dy xdx
52
x x
3If 3, find 2viii f x x
f
3 3f x x
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
21 3vi y xx
23x x 33 2dy x
dx
323x
2 vii y x x52x
325
2dy xdx
52
x x
3If 3, find 2viii f x x
f
3 3f x x
23f x x
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
21 3vi y xx
23x x 33 2dy x
dx
323x
2 vii y x x52x
325
2dy xdx
52
x x
3If 3, find 2viii f x x
f
3 3f x x
23f x x
22 3 2f
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
21 3vi y xx
23x x 33 2dy x
dx
323x
2 vii y x x52x
325
2dy xdx
52
x x
3If 3, find 2viii f x x
f
3 3f x x
23f x x
22 3 2f
12
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
2when 1, 15 1 12 1dyxdx
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
2when 1, 15 1 12 1dyxdx
3
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
2when 1, 15 1 12 1dyxdx
3required slope 3
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
2when 1, 15 1 12 1dyxdx
3required slope 3
1 3 1y x
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
2when 1, 15 1 12 1dyxdx
3required slope 3
1 3 1y x
1 3 3y x
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
2when 1, 15 1 12 1dyxdx
3required slope 3
1 3 1y x
1 3 3y x
3 2 0x y
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
2when 1, 15 1 12 1dyxdx
3required slope 3
1 3 1y x
1 3 3y x
3 2 0x y
3 Find the points on the curve 12 where the tangents are horizontalx y x x
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
2when 1, 15 1 12 1dyxdx
3required slope 3
1 3 1y x
1 3 3y x
3 2 0x y
3 Find the points on the curve 12 where the tangents are horizontalx y x x
3 12y x x
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
2when 1, 15 1 12 1dyxdx
3required slope 3
1 3 1y x
1 3 3y x
3 2 0x y
3 Find the points on the curve 12 where the tangents are horizontalx y x x
3 12y x x
23 12dy xdx
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
2when 1, 15 1 12 1dyxdx
3required slope 3
1 3 1y x
1 3 3y x
3 2 0x y
3 Find the points on the curve 12 where the tangents are horizontalx y x x
3 12y x x
23 12dy xdx
tangents are horizontal when 0dydx
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
2when 1, 15 1 12 1dyxdx
3required slope 3
1 3 1y x
1 3 3y x
3 2 0x y
3 Find the points on the curve 12 where the tangents are horizontalx y x x
3 12y x x
23 12dy xdx
tangents are horizontal when 0dydx
2i.e. 3 12 0x
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
2when 1, 15 1 12 1dyxdx
3required slope 3
1 3 1y x
1 3 3y x
3 2 0x y
3 Find the points on the curve 12 where the tangents are horizontalx y x x
3 12y x x
23 12dy xdx
tangents are horizontal when 0dydx
2i.e. 3 12 0x
2 42
xx
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
2when 1, 15 1 12 1dyxdx
3required slope 3
1 3 1y x
1 3 3y x
3 2 0x y
3 Find the points on the curve 12 where the tangents are horizontalx y x x
3 12y x x
23 12dy xdx
tangents are horizontal when 0dydx
2i.e. 3 12 0x
2 42
xx
tangents are horizontal at 2,16 and 2, 16
A normal is a line perpendicular to the tangent at the point of contact
A normal is a line perpendicular to the tangent at the point of contacty
x
y f x
A normal is a line perpendicular to the tangent at the point of contacty
x
y f x
tangent
A normal is a line perpendicular to the tangent at the point of contacty
x
y f x
tangentnormal
A normal is a line perpendicular to the tangent at the point of contacty
x
y f x
tangentnormal
2 Find the equation of the normal to the curve 4 3 2 at the point 3,29xi y x x
A normal is a line perpendicular to the tangent at the point of contacty
x
y f x
tangentnormal
2 Find the equation of the normal to the curve 4 3 2 at the point 3,29xi y x x
24 3 2y x x
A normal is a line perpendicular to the tangent at the point of contacty
x
y f x
tangentnormal
2 Find the equation of the normal to the curve 4 3 2 at the point 3,29xi y x x
24 3 2y x x
8 3dy xdx
A normal is a line perpendicular to the tangent at the point of contacty
x
y f x
tangentnormal
2 Find the equation of the normal to the curve 4 3 2 at the point 3,29xi y x x
24 3 2y x x
8 3dy xdx
when 3, 8 3 3dyxdx
A normal is a line perpendicular to the tangent at the point of contacty
x
y f x
tangentnormal
2 Find the equation of the normal to the curve 4 3 2 at the point 3,29xi y x x
24 3 2y x x
8 3dy xdx
when 3, 8 3 3dyxdx
21
A normal is a line perpendicular to the tangent at the point of contacty
x
y f x
tangentnormal
2 Find the equation of the normal to the curve 4 3 2 at the point 3,29xi y x x
24 3 2y x x
8 3dy xdx
when 3, 8 3 3dyxdx
21 1 required slope21
A normal is a line perpendicular to the tangent at the point of contacty
x
y f x
tangentnormal
2 Find the equation of the normal to the curve 4 3 2 at the point 3,29xi y x x
24 3 2y x x
8 3dy xdx
when 3, 8 3 3dyxdx
21 1 required slope21
129 321
y x
A normal is a line perpendicular to the tangent at the point of contacty
x
y f x
tangentnormal
2 Find the equation of the normal to the curve 4 3 2 at the point 3,29xi y x x
24 3 2y x x
8 3dy xdx
when 3, 8 3 3dyxdx
21 1 required slope21
129 321
y x
21 609 3y x
A normal is a line perpendicular to the tangent at the point of contacty
x
y f x
tangentnormal
2 Find the equation of the normal to the curve 4 3 2 at the point 3,29xi y x x
24 3 2y x x
8 3dy xdx
when 3, 8 3 3dyxdx
21 1 required slope21
129 321
y x
21 609 3y x
21 612 0x y
Exercise 7C; 1ace etc, 2ace etc, 3ace etc, 4bd, 5bdfh, 8bd, 9bd, 10ac, 12, 13b, 16, 21
Exercise 7D; 2ac, 3bd, 4ace, 6c, 7b, 11a, 13aei, 18, 22
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