Step by step
The question
Question: find the derivative of f(x) = log₂(x² + 4).
f(x) = log2(x2 + 4)
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 + 4, over ln 2.
ln 2 is just a number.
log2(x2 + 4) = ln(x2 + 4)ln 2
2 The rule
The number 1 over ln 2 waits.
ln u gives 1/u, times the inside's slope, 2x.
ddxln(x2 + 4)ln 2 = 1ln 2 × 1(x2 + 4) × 2x
Careful!
That is only true for ln.
For log base 2, divide by ln 2 too.
Careful: not log2(x) → 1x ✗
3 Tidy up
Multiply into one fraction: f′(x) = 2x over ((x² + 4) ln 2).
f′(x) = 2x(x2 + 4) ln 2
4 Check
Let's check: the derivative is the slope.
At x = 1, our answer gives 0.5771, with a calculator.
f′(1) = 0.5771
Now the slope between two very close points: x = 1 and x = 1.001.
A calculator gives about 0.58.
Very close to 0.5771.
Right!
f(1.001) − f(1)0.001 ≈ 0.58
Let's look back.
First: write it with e.
Then: the rule.
Then: tidy up.
Last: check.
Answer
f′(x) = 2x(x2 + 4) ln 2