Derivative of 3e^(x^2 + 1)

Step by step

The question

Question: find dy/dx for y = 3e^(x² + 1).

1 The inside

There is something inside: x² + 1.

Call it u.

Then y is just 3eu.

2 The outside

eˣ is special: its derivative is itself.

Its slope always equals its height.

First the outside.

The rule on 3eu gives 3eu.

3 The inside's slope

Now the inside.

The power rule on x² + 1 gives 2x.

4 Multiply

The chain rule: multiply the two.

dy/du times du/dx.

Put the inside back in place of u.

Careful!

Do not stop here!

We must multiply by the inside's slope, 2x.

Write the numbers in front.

So dy/dx = 6x e^(x² + 1).

5 Check

Let's check: the derivative is the slope.

At x = 1, the rule gives 44.3343, with a calculator.

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

A calculator gives about 44.4.

Very close to 44.3343.

Right!

Let's look back.

First: the inside.

Then: the outside.

Then: the inside's slope.

Then: multiply.

Last: check.