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

Step by step

The question

Question: find dy/dx for y = e⁻ˣ.

y = e^(−x)

There are four choices.

First we solve it, the normal way.

Then we pick.

1 The inside

There is something inside: −x.

Call it u.

u = −x

Then y is just e to the u.

y = e^u

2 The outside

eˣ is special: its derivative is itself.

Its slope always equals its height.

e^x → e^x

First the outside.

The rule on e to the u gives e to the u.

dydu = e^u

3 The inside's slope

Now the inside.

The power rule on −x gives −1.

dudx = −1

4 Multiply

The chain rule: the outside's derivative times the inside's derivative, dy/du times du/dx.

dydx = dydu × dudx

Put the inside back in place of u.

= e^(−x) × −1

Careful!

Do not stop here!

We must multiply by the inside's slope, −1.

Careful: not e^(−x) ✗

Write the numbers in front.

So dy/dx = −e⁻ˣ.

= −e^(−x)

5 Check

Let's check: the derivative is the slope.

At x = 1, the rule gives −0.3679, with a calculator.

dy/dx at x = 1: −0.3679

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

A calculator gives about −0.37.

Very close to −0.3679.

Right!

y(1.001) − y(1)0.001 ≈ −0.37

6 Pick the choice

Our answer is dy over dx = −e⁻ˣ.

That is choice C.

C) −e^−x

7 Why not the others

A is e⁻ˣ.

Wrong: lost the −1, the derivative of the inside.

That is the Careful mistake from before.

A) e^−x ✗ lost the −1, the derivative of the inside

B is −xe^(−x − 1).

Wrong: eᵘ is not a power of x.

B) −xe^{−x − 1} ✗ e^u is not a power of x

D is −eˣ.

Wrong: the power keeps its minus.

D) −e^x ✗ the power keeps its minus

Let's look back.

First: the inside.

Then: the outside.

Then: the inside's slope.

Then: multiply.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer