Step by step
The question
Question: find the derivative of f(x) = log(5x + 1).
f(x) = log(5x + 1)
1 Write it with e
Change of base: log base b of x is ln x over ln b.
log with no base is base 10.
logb(x) = ln xln b
So it is ln of 5x + 1, over ln 10.
ln 10 is just a number.
log(5x + 1) = ln(5x + 1)ln 10
2 The rule
The number 1 over ln 10 waits.
ln u gives 1/u, times the inside's slope, 5.
ddxln(5x + 1)ln 10 = 1ln 10 × 1(5x + 1) × 5
Careful!
That is only true for ln.
For log base 10, divide by ln 10 too.
Careful: not log(x) → 1x ✗
3 Tidy up
Multiply into one fraction: f′(x) = 5 over ((5x + 1) ln 10).
f′(x) = 5(5x + 1) ln 10
4 Check
Let's check: the derivative is the slope.
At x = 1, our answer gives 0.3619, with a calculator.
f′(1) = 0.3619
Now the slope between two very close points: x = 1 and x = 1.001.
A calculator gives about 0.36.
Very close to 0.3619.
Right!
f(1.001) − f(1)0.001 ≈ 0.36
Let's look back.
First: write it with e.
Then: the rule.
Then: tidy up.
Last: check.
Answer
f′(x) = 5(5x + 1) ln 10