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Differentiation I
The Definition of Differentiation
...
The derivative is the
instantaneous rate of change of a function with respect to one of its variables
...
For example, we may have a function
like 𝑌 = 𝑥 3 + 4𝑥 + 3, Here we can’t just simply state the gradient as we can with equations in the
form 𝑌 = 𝑚𝑥 + 𝑐
...
e
...
This is simple enough, but it can become more
complicated
...
e
...
So, the chain rule must be applied:
This can be put in the form 𝑓(𝑥) = 𝑆𝑖𝑛(𝑢) where 𝑢 = 𝑥 2 from this both 𝑓(𝑥) 𝑎𝑛𝑑 𝑢 need
to be differentiated then multiplied together
...
e
...
For example, 𝑓(𝑥) = 𝑥𝑠𝑖𝑛(𝑥)
to solve this requires 3 steps (This is the
product rule)
Step1 – split into two variables usually u= and v=
Step2 – differentiate both u and v with respect to x
Robert Airey

Step 3- multiply diagonally and add
If we take the example of 𝑓(𝑥) = 𝑥𝑠𝑖𝑛(𝑥)
Step 1𝑢 = 𝑥 𝑎𝑛𝑑 𝑣 = sin(𝑥)

the order doesn’t matter

Step 2 and 3 –

𝑢=𝑥

𝑣 = sin(𝑥)

𝑢′ = 1

𝑣′ = cos(𝑥)

The product rule is 𝑓(𝑥) = 𝑢(𝑥)𝑣(𝑥) 𝑡ℎ𝑒𝑛 𝑓 ′ (𝑥) = 𝑣𝑢′ + 𝑢𝑣 ′
Again, the order of this doesn’t matter
𝑢(𝑥)

Now let’s look at fractions in the form 𝑓(𝑥) = 𝑣(𝑥)

the order of this does matter, the

numerator should always be u(x) and denominator v(x), this is the quotient rule
Again 3 steps
Step1- state 𝒖(𝒙) =

𝒗(𝒙) =

Step2- differentiate 𝒖(𝒙) 𝒂𝒏𝒅 𝒗(𝒙)
Step3- diagonally multiply (like for the product rule) except subtract and divide by (𝒗(𝒙))𝟐
3𝑥 2

For example, let’s take 𝑓(𝑥) = cos(𝑥)
𝑢(𝑥)

Put this in the form 𝑓(𝑥) = 𝑣(𝑥)

𝑢(𝑥 ) = 3𝑥 2

𝑣(𝑥 ) = cos(𝑥)

Find the derivatives

𝑢(𝑥 ) = 3𝑥 2

𝑣(𝑥 ) = cos(𝑥)

𝑢′ (𝑥 ) = 6𝑥

𝑣 ′ (𝑥) = −sin(𝑥)

So 𝑓(𝑥) =
So 𝑓(𝑥) =

𝑢(𝑥)
𝑣(𝑥)

3𝑥 2
cos(𝑥)

Robert Airey

𝑓 ′ (𝑥) =

𝑣𝑢′ −𝑢𝑣 ′
𝑣2

𝑓 ′ (𝑥) =

(6𝑥𝑐𝑜𝑠(𝑥)−−3𝑥 2 sin(𝑥))
cos2(𝑥)

=

6𝑥𝑐𝑜𝑠(𝑥)+3𝑥 2 sin(𝑥)
cos2 (𝑥)

Natural logs and exponentials are easier to differentiate if we remember the rules,
𝑓 ′ (𝑥) =

𝑓(𝑥) = ln(𝑥)
𝑓(𝑥) = 𝑒 𝑥

1
𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑙 𝑜𝑓 𝑏𝑟𝑎𝑐𝑘𝑒𝑡
𝑖𝑛 𝑙𝑎𝑦𝑚𝑎𝑛𝑠 𝑡𝑒𝑟𝑚𝑠 𝑖𝑡 ′ 𝑠
𝑥
𝑏𝑟𝑎𝑐𝑘𝑒𝑡
2

𝑓 ′ (𝑥) = 𝑒 𝑥 , 𝑖𝑓 𝑖𝑡 𝑤𝑒𝑟𝑒 𝑓(𝑥) = 𝑒 𝑥 𝑡ℎ𝑒𝑛 𝑓 ′ (𝑥) = 2𝑥𝑒 𝑥

exponentials it’s easier to generalise as 𝑓(𝑥) = 𝑒 𝑢(𝑥) 𝑡ℎ𝑒𝑛 𝑓 ′ (𝑥) =

𝑑𝑢(𝑥)
𝑑𝑥

2

so, for

𝑒 𝑢(𝑥)

Logs are especially useful to differentiate when equations like this come along𝑦 = 3𝑥 2 × sin(𝑥) × cos2 (𝑥)
It is possible to product rule this twice however by natural logging both sides we get
ln(𝑦) = ln(3𝑥 2 ) × ln(sin(𝑥)) × ln(cos2 (𝑥))
Simplifying using log laws we get
ln(𝑦) = 2 ln(3𝑥) + ln(𝑠𝑖𝑛(𝑥)) + 2ln(𝑐𝑜𝑠(𝑥))
Differentiation gives us
1 𝑑𝑦
6 cos(𝑥) 2 sin(𝑥)
=
+
−
𝑦 𝑑𝑥 3𝑥 sin(𝑥)
cos(𝑥)
𝑑𝑦

2

cos(𝑥)

Getting 𝑑𝑥 = (𝑦) × (𝑥 + sin(𝑥) − 2 tan(𝑥))
Substitution for y if needed in terms of x only

Second derivatives are also useful
...

For example, take 𝑦 = 3𝑥 2 + 4𝑥
𝑑𝑦

We differentiate as normal therefore giving us 𝑑𝑥 = 6𝑥 + 4
As we are looking for where the gradient is zero, we set
𝑑𝑦
𝑑𝑥

= 0 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝑡ℎ𝑖𝑠 𝑏𝑒𝑐𝑜𝑚𝑒𝑠 6𝑥 + 4 = 0
4

4

Therefore at 𝑥 = − 6 𝑎𝑛𝑑 𝑓𝑖𝑛𝑑𝑖𝑛𝑔 𝑦, 𝑦 = − 3 at this point there is either a maximum or a
minimum point
To determine this points nature, we must differentiate
𝑑𝑦
𝑑𝑥

𝑑2 𝑦

= 6𝑥 + 4 𝑎𝑔𝑎𝑖𝑛 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝑑𝑥 2 = 6

Robert Airey

𝑑2 𝑦

𝑑2 𝑦

If 𝑑𝑥 2 > 0 𝑡ℎ𝑖𝑠 𝑖𝑠 𝑎 𝑚𝑖𝑛𝑖𝑚𝑢𝑚, 𝑖𝑓 𝑑𝑥 2 < 0 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑝𝑜𝑖𝑛𝑡 think of this as, a minimum
point as having a gradient that starts to increase hence positive and a maximum point
having a gradient that starts to decrease hence negative

Standard differentials

𝑓(x) = tan(kx) − − − − − 𝑓′(𝑥) = 𝑘𝑠𝑒𝑐 2 (𝑘𝑥)
𝑓(𝑥 ) = sec(𝑥 ) − − − − − 𝑓 ′ (𝑥) = sec(𝑥 ) tan(𝑥 )
𝑓(𝑥 ) = cot(𝑥 ) − − − − − 𝑓 ′ (𝑥 ) = −𝑐𝑜𝑠𝑒𝑐 2 (𝑥 )
𝑓 (𝑥 ) = 𝑐𝑜𝑠𝑒𝑐(𝑥 ) − − − −𝑓 ′ (𝑥 ) = −𝑐𝑜𝑠𝑒𝑐(𝑥 )cot(𝑥)

Robert Airey

Questions
Q1- if 𝑓(𝑥) = 𝑥 3 + 2𝑥 + ln(𝑥) ,state the derivative at x=2
Q2- Find the derivative of 𝑓(𝑥) = cos(𝑥 2 )
Q3- Find the derivative of 𝑓(𝑥) = sin(𝑥) cos(𝑥)
Q4- Find the derivative of 𝑓(𝑥) =

ln(𝑥)
4𝑥 2

Q5- Find the coordinates of the stationary point for 𝑦 = −𝑒 2𝑥 + 4𝑥 and
determine its nature
Q6- Find the derivative of 𝑦 = sin2 (𝑥) cos(x) e2x
Q7- Find the derivative of 𝑓(𝑥) =

sin(𝑥 2 )
ln(𝑥)

Q8- Find the derivative of 𝑦 = 𝑥 3 + 2𝑥 + 4 at x=10
Q9- Find the derivative of 𝑓(𝑥) = ln(𝑥) 𝑒 𝑥

2

Q10- 𝑓(𝑥) = 𝑥 3 + 3𝑥 2 + 1 find the derivative and any coordinates of
stationary points

Robert Airey

Solutions
Q1) apply general power rule and differential rule of natural logs
𝑑𝑦
𝑑𝑥

= 3𝑥 2 + 2 +

1
𝑥

substitute x=2 then

𝑑𝑦
𝑑𝑥

= 14

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