Derivative of log_5(x^2)

Step by step

The question

Question: find the derivative of f(x) = log₅(x²).

1 Write it with e

Change of base: log base b of x is ln x over ln b.

So it is ln of x², over ln 5.

ln 5 is just a number.

2 The rule

The number 1 over ln 5 waits.

ln u gives 1/u, times the inside's slope, 2x.

Careful!

That is only true for ln.

For log base 5, divide by ln 5 too.

3 Tidy up

Multiply into one fraction: f′(x) = 2x/(x² ln 5).

4 Check

Let's check: the derivative is the slope.

At x = 1, our answer gives 1.2427, with a calculator.

Now the slope between two very close points: x = 1 and x = 1.001.

A calculator gives about 1.24.

Very close to 1.2427.

Right!

Let's look back.

First: write it with e.

Then: the rule.

Then: tidy up.

Last: check.