Derivative of ln⁡(𝑥)

Let's work some practice problems taking the derivative of the natural log, ln⁡(𝑥), each of course with a complete solution a click away.

Note that problems involving this particular derivative essentially always require the Chain Rule, so this is good additional Chain Rule practice too.

Matheno Essentials: Derivative of ln⁡(𝑥)

𝑑𝑑𝑥ln⁡𝑥=1𝑥

Applying the Chain rule, we have

𝑑𝑑𝑥ln⁡(stuff)=1(stuff)⋅𝑑𝑑𝑥(stuff)

You'll usually see this written as

𝑑𝑑𝑥ln⁡𝑢=1𝑢⋅𝑑𝑢𝑑𝑥
Practice Problem #1
Differentiate 𝑓(𝑥) =ln⁡(𝑥2 +3𝑥 −1).
Practice Problem #2
Differentiate 𝑓(𝑥) =√ln⁡𝑥2.
Practice Problem #3
Differentiate 𝑓(𝑥) =ln⁡(ln⁡𝑥).
Practice Problem #4
Differentiate 𝑓(𝑥) =ln⁡(𝑥2𝑒𝑥).
Practice Problem #5
Differentiate 𝑓(𝑥) =ln⁡(cos⁡𝑥5𝑥).

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