Step by step
The question
Question: find the derivative of f(x) = 4eˣ − 3 ln x.
f(x) = 4ex − 3 ln x
1 The rules
eˣ is special: its derivative is itself.
Its slope always equals its height.
ex → ex
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) = 4 × ex − 3 × 1x
Careful!
The power rule is only for x to a number.
In e to the x, the x is the power.
So e to the x stays e to the x.
Careful: not ex → xex − 1 ✗
3 Tidy up
So f′(x) = 4 eˣ − 3 over x.
= 4ex − 3x
4 Check
Let's check: the derivative is the slope.
At x = 1, the rule gives 7.8731, with a calculator in radians.
f′(1) = 4e − 3 ≈ 7.8731
Now the slope between two very close points: x = 1 and x = 1.001.
A calculator gives about 7.88.
Very close to 7.8731.
Right!
f(1.001) − f(1)0.001 ≈ 7.88
Let's look back.
First: the rules.
Then: each term.
Then: tidy up.
Last: check.
Answer
= 4ex − 3x