If f(x) = 6e^x + 2ln x, what is f′(x)? Step by Step

Step by step

The question

Question: find the derivative of f(x) = 6eˣ + 2 ln x.

f(x) = 6e^x + 2 ln x

There are four choices.

First we solve it, the normal way.

Then we pick.

1 The rules

eˣ is special: its derivative is itself.

Its slope always equals its height.

e^x → e^x

The derivative of ln x is 1 over x.

ln x → 1x

2 Each term

Use the rule on each term.

A number in front stays, and multiplies.

f′(x) = 6 × e^x + 2 × 1x

Careful!

The power rule is only for x to a number.

In e to the x, the x is the power.

So e to the x stays e to the x.

Careful: not e^x → xe^{x − 1} ✗

3 Tidy up

So f′(x) = 6 eˣ + 2 over x.

= 6e^x + 2x

4 Check

Let's check: the derivative is the slope.

At x = 1, the rule gives 18.3097, with a calculator in radians.

f′(1) = 6e + 2 ≈ 18.3097

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

A calculator gives about 18.32.

Very close to 18.3097.

Right!

f(1.001) − f(1)0.001 ≈ 18.32

5 Pick the choice

Our answer is f′(x) = 6eˣ + 2 over x.

That is choice B.

B) 6e^x + 2x

6 Why not the others

A is 6eˣ + 2x.

Wrong: ln x gives 1/x, not x.

A) 6e^x + 2x ✗ ln x gives 1/x, not x

C is 6xe^(x − 1) + 2/x.

Wrong: used the power rule on eˣ: it stays eˣ.

C) 6xe^{x − 1} + 2x ✗ used the power rule on e^x: it stays e^x

D is 6eˣ + 1/(2x).

Wrong: 2ln x gives 2/x, not 1/(2x).

D) 6e^x + 12x ✗ 2ln x gives 2/x, not 1/(2x)

Let's look back.

First: the rules.

Then: each term.

Then: tidy up.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer