Step by step
The question
Question: find the derivative of f(x) = log₃(x² + 1).
f(x) = log3(x2 + 1)
1 Write it with e
Change of base: log base b of x is ln x over ln b.
logb(x) = ln xln b
So it is ln of x² + 1, over ln 3.
ln 3 is just a number.
log3(x2 + 1) = ln(x2 + 1)ln 3
2 The rule
The number 1 over ln 3 waits.
ln u gives 1/u, times the inside's slope, 2x.
ddxln(x2 + 1)ln 3 = 1ln 3 × 1(x2 + 1) × 2x
Careful!
That is only true for ln.
For log base 3, divide by ln 3 too.
Careful: not log3(x) → 1x ✗
3 Tidy up
Multiply into one fraction: f′(x) = 2x over ((x² + 1) ln 3).
f′(x) = 2x(x2 + 1) ln 3
4 Check
Let's check: the derivative is the slope.
At x = 1, our answer gives 0.9102, with a calculator.
f′(1) = 0.9102
Now the slope between two very close points: x = 1 and x = 1.001.
A calculator gives about 0.91.
Very close to 0.9102.
Right!
f(1.001) − f(1)0.001 ≈ 0.91
Let's look back.
First: write it with e.
Then: the rule.
Then: tidy up.
Last: check.
Answer
f′(x) = 2x(x2 + 1) ln 3