VIDEO SUMMARIES: DIFFERENTIATION *
What*you*need*to*know:!
•  The*gradient*at*a*point*
Differen(ate!an!equa(on!to!get!
•  Differen'ate,+then+subs'tute+x+value+
a!gradient!
•  Sta>onary*points*
! •  Differen'ate,+set+f’(x)+=+0,+then+solve+
We!do!this!by:!
•  Maxima*or*Minima*
1. 
Mul(plying!the!power!by!
•  2nd+differen'al,+Maxima+(>),+Minima+(+)+
the!coefficient!
•  Gradient*of*a*tangent*line*
2. 
Subtrac(ng!1!from!the!
•  Same+as+gradient+at+a+point+
power!
•  Gradient*of*a*normal*line*
•  Nega've+reciprocal+of+tangent+(>1/m)+
Steps%(frac5on%!%root)!
Steps%(root%!%frac5on)!
1.  Draw!root!sign!
1.  Keep!power!as!top!of!
2.  Move!bo3om!
frac@on!
(denominator)!in!front!of! 2.  Put!root!number!on!
root!
bo3om!of!frac@on!
3.  Leave!top!(numerator)!
3.  Get!rid!of!root!symbol!
as!power!of!x!
Note:%if!no!root!number!
denominator!=2!
DIFFERENTIATION*OVERVIEW
ROOTS%AND%EXPONENTS%
What%you%need%to%know%
n
d
d
x = x
n
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VIDEO SUMMARIES: DIFFERENTIATION FRACTIONS*AND*POWERS*
A"denominator"can"become"a"numerator,"if"you"make"the"powers"nega8ve"
2
−3
5⋅ (x − 5)
5
=
2
3
17
17 ⋅ (x − 5)
GRAPHS'
LeJ)&)right)
go)to)same)
point)
What)you)need)to)know)
Not))
smooth)&)
con%nuous)
Flat)
f%‘(x)%=)0)
Part'1:'Equa%on,)f(x)%
Part'3:'Concavity,)f%‘‘(x):)
)a))read)graph)values)
)f))concave)up)>0)
)b))lim)f(x))as)x)!)a)
)g))concave)down)<0)
)c))not)differen%able)
)h))point)of)inflec%on)=0)
)
)
Part'2:'Differen%al)f%‘(x))
)d))not)defined)
)e))sta%onary)points)
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VIDEO SUMMARIES: DIFFERENTIATION *
Natural'long'(ln)'
Euler’s'number'(e)'
Trig'func7ons'(sin…)'
What*you*need*to*know:*
Use'differen7a7on'table'to'differen7ate'func7ons'
f (x) = sin x + cos x
f (x) = ln x
f (x) = e 2 x
1
f '(x) = cos x − sin x
f '(x) =
f '(x) = 2e 2 x
x
&
For$when$one$differen,able$term$is$inside$another$differen,able$term$
Steps&
For$example:$
1.  Iden,fy$inner$and$outer$terms$
2.  Differen,ate$inner$and$outer$
3(x 2 − 2x)3
3.  Mul,ply$inner$’$and$outer$’$
4.  Subs,tute$inner$
Inner$
Outer$
3
3(inner)
Inner$’$x$Outer$’$$
x 2 − 2x
2
Outer$’$
Inner$’$
(2x
−
2)⋅
9(inner)
9(inner)2
2x − 2
Subs,tute$inner$
(2x − 2)⋅ 9(x 2 − 2x)2
DIFFERENTIATION*SKILLS
CHAIN&RULE
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VIDEO SUMMARIES: DIFFERENTIATION PRODUCT(RULE(
For$when$you$have$2$differen1able$terms$mul1plied$by$each$other$
For$example:$
2
Product$Rule:$
3
3x (x + 2)
f$
3x
1.  Differen1ate$both$terms$
2.  Put$into$equa1on$
3
2
Steps(
g$
x +2
3x 2 .3x 2 + 6x(x 3 + 2)
g$’$
2
f$’$
3x
6x
9x 4 + 6x 4 +12x
QUOTIENT(RULE(
For$example:$
For$when$you$have$2$differen1able$terms$in$a$frac1on$
Quo1ent$Rule:$
3
3x + 2
x5
f$
3
3x + 2
f$’$
9x
2
Steps(
1.  Differen1ate$both$terms$
2.  Put$into$equa1on$
g$
5
x
g$’$
4
5x
'
⎛ f ⎞ x 5.9x 2 − (3x 3 + 2)5x 4 9x 7 −15x 7 −10x 4
=
⎜ ⎟=
10
5 2
x
⎝g⎠
x
( )
−6x 7 −10x 4 −6x 3 −10
=
=
10
x6
x
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VIDEO SUMMARIES: DIFFERENTIATION )
For$when$x$and$y are$explained$by$a$3rd$variable$
For$example:$
Steps)
dy Parametric$Func9on$
2
1. 
Differen9ate$both$func9ons$with$
y = 3t x = 6t
=?
respect$to$the$3rd$variable$
dx
2.  Flip$one$of$the$answers$
3.  Mul9ply$together$
dx
dt 1
dy =6
=
= 6t
1
6t
dt
dx
6
dt
=t
6
t
x
=
6
6
&
Steps&
r = 1.5t 2
The*radius*increases*according*to*the*formula:*
1. 
Iden'fy*the*rate*you*want*to*find*
At*what*rate*is*the*area*inside*the*ripple*increasing**
2.  Iden'fy*the*rates*you*have*been*given*
aHer*2*sec?*
dA
2 3.  Use*the*parametric*equa'on*to**
A
=
r
dA
dr
=2 r
find*what*is*missing*
=
3
t
dr
4.  Find*missing*differen'al**
dt dt
or*repeat*as*above.*
dA
2 2
2
= 3t x r = 3t x (1.5t ) 5.  Subs'tute*and*solve*
dA
dr dA
dt
=
x
dA
dt dt dr At*2*sec*
= 3(2) (1.5(2) 2 ) 2 = 678.6 cm2/s*
dt
PARAMETRIC)EQUATIONS
EXCELLENCE&QUESTIONS&–&PARAMETRIC
π
π
π
π
π
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VIDEO SUMMARIES: DIFFERENTIATION EXCELLENCE&QUESTIONS&–&MAXIMUMS&
Steps&
A*cylinder*of*height*h*cm*and*radius*r*cm*is*inscribed*inside*
1.  Iden'fy*what*you*need*
a*sphere*of*radius*10*cm.*
2.  Write*equa'on*that*includes*all*variables*
Find*the*value*of*r*that*maximises*the*volume*of*the*
3.  Get*in*terms*of*one*variable*
cylinder.*
4.  Differen'ate*and*set*equal*to*0*
You*do*not*need*to*prove*that*the*volume*is*a*maximum*
5.  Solve*for*variable*
and*not*a*minimum.*
6. 
If*it*is*not*the*answer*use**
dV
V = πr 2 h h = 400 − 4r 2 solu'on*to*solve*for*other*answer**
dr
20
2r
h
V = πr 2 400 − 4r 2
4π r 3
2
dV
= 2π r. 400 − 4r −
=0
dr
400 − 4r 2
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r=
200
= 8.16 cm*
3