Derivative of 5^(3x)

Step by step

The question

Question: find the derivative of f(x) = 5^(3x).

1 Write it with e

We know how to take the derivative of e to a power.

And any number is e to a power: 5 is e to the ln 5.

So 5 to the 3x is e to the 3x times ln 5.

2 The rule

The derivative of e to a power is itself, times the slope of the power: 3 times ln 5.

The rule: b to the x gives b to the x, times ln b.

Careful!

The power rule is only for x to a fixed number.

Here x is in the power: use b to the x, times ln b.

3 Tidy up

Put it back in base 5, with the numbers in front: f′(x) = 3 × 5^(3x) ln 5.

4 Check

Let's check: the derivative is the slope.

At x = 1, our answer gives 603.5392, with a calculator.

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

A calculator gives about 605.

Very close to 603.5392.

Right!

Let's look back.

First: write it with e.

Then: the rule.

Then: tidy up.

Last: check.