If: 𝑦 = 𝑒 𝑥
If: 𝑦 = 𝑒 𝑓(𝑥)
Derivatives of Logarithmic Functions
Then:
𝑑𝑦
𝑑𝑥
Then:
= 𝑒𝑥
𝑑𝑦
𝑑𝑥
= 𝑓′(𝑥)𝑒 𝑓(𝑥)
If: 𝑦 = 𝑙𝑛𝑥
Then:
Let: 𝑦 = 𝑙𝑛𝑥
𝑑𝑦
𝑑𝑥
=
1
𝑥
Inverse Logarithm
𝑥 = 𝑒𝑦
𝑑𝑥
= 𝑒𝑦
𝑑𝑦
𝑑𝑦
1
= 𝑦
𝑑𝑥 𝑒
𝑑𝑦 1
=
𝑑𝑥 𝑥
Differentiate
𝑑𝑦
1
=
𝑑𝑥
𝑑𝑥
𝑑𝑦
Earlier we said
that 𝑥 = 𝑒 𝑦
9B
If: 𝑦 = 𝑒 𝑥
If: 𝑦 = 𝑒 𝑓(𝑥)
Derivatives of Logarithmic Functions
Then:
𝑑𝑦
𝑑𝑥
Then:
= 𝑒𝑥
𝑑𝑦
𝑑𝑥
= 𝑓′(𝑥)𝑒 𝑓(𝑥)
If: 𝑦 = 𝑙𝑛𝑥
Then:
𝑦 = 𝑎𝑥
𝑙𝑛𝑦 = 𝑙𝑛𝑎 𝑥
𝑙𝑛𝑦 = 𝑥𝑙𝑛𝑎
𝑦=𝑒
𝑥𝑙𝑛𝑎
𝑑𝑦
𝑑𝑥
=
1
𝑥
Take natural logs of both sides
Use the power law
Rewrite using inverse logs
Differentiate using the rule above
(remember 𝑙𝑛𝑎 is a constant)
𝑑𝑦
= (𝑙𝑛𝑎)𝑒 𝑥𝑙𝑛𝑎
𝑑𝑥
We can replace 𝑒 𝑥 𝑙𝑛𝑎 using a previous step
𝑑𝑦
= 𝑙𝑛𝑎 𝑦
𝑑𝑥
We can then replace 𝑦 using a previous step
𝑑𝑦
= 𝑙𝑛𝑎 𝑎 𝑥
𝑑𝑥
Rewrite
𝑑𝑦
= 𝑎 𝑥 𝑙𝑛𝑎
𝑑𝑥
9B