If f(x) = 5^x, what is f′(x)? Step by Step

Step by step

The question

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

f(x) = 5ˣ

There are four choices.

First we solve it, the normal way.

Then we pick.

1 The rule

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

bˣ → bˣ 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.

Careful: not 5ˣ → x5^(x − 1) ✗

2 Use the rule

5 to the x gives 5 to the x, times ln 5.

So f′(x) = 5ˣ ln 5.

f′(x) = 5ˣ ln 5

3 Check

Let's check at one x.

Any x works: take x = 1.

The derivative is the slope there.

Our answer gives 8.0472, with a calculator.

f′(1) = 8.0472

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

A calculator gives about 8.05.

Very close to 8.0472.

Right!

f(1.001) − f(1)0.001 ≈ 8.05

4 Pick the choice

Our answer is f′(x) = 5ˣ ln 5.

That is choice C.

C) 5^x ln 5

5 Why not the others

A is x × 5^(x − 1).

Wrong: the power rule needs a number power.

A) x × 5^{x − 1} ✗ the power rule needs a number power

B is 5ˣ.

Wrong: only eˣ is its own derivative.

B) 5^x ✗ only e^x is its own derivative

D is 5ˣ/ln 5.

Wrong: multiply by ln 5, do not divide.

D) 5^x/ln 5 ✗ multiply by ln 5, do not divide

Let's look back.

First: the rule.

Then: use the rule.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer