Step by step
The question
Question: find the derivative of f(x) = 5 ln x.
f(x) = 5 ln x
1 The rules
The derivative of ln x is 1 over x.
ln x → 1x
2 Each term
Use the rule on each term.
A number in front just waits, and multiplies.
f′(x) = 5 × 1x
3 Tidy up
So f′(x) = 5 over x.
= 5x
4 Check
Let's check: the derivative is the slope.
At x = 1, the rule gives 5, with a calculator in radians.
f′(1) = 5 ≈ 5
Now the slope between two very close points: x = 1 and x = 1.001.
A calculator gives about 5.
Very close to 5.
Right!
f(1.001) − f(1)0.001 ≈ 5
Let's look back.
First: the rules.
Then: each term.
Then: tidy up.
Last: check.
Answer
= 5x