Derivative of e^(x^2) Step by Step | Derivatives of sin, cos, e and ln

Step by step

The question

Question: find dy/dx for y = e^(x²).

y = e^(x²)

1 The inside

There is something inside: x².

Call it u.

u = x²

Then y is just eu.

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 eu gives eu.

dydu = e^u

3 The inside's slope

Now the inside.

The power rule on x² gives 2x.

dudx = 2x

4 Multiply

The chain rule: multiply the two.

dy/du times du/dx.

dydx = dydu × dudx

Put the inside back in place of u.

= e^(x²) × 2x

Careful!

Do not stop here!

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

Careful: not e^(x²) ✗

Write the numbers in front.

So dy/dx = 2x e^(x²).

= 2x e^(x²)

5 Check

Let's check: the derivative is the slope.

At x = 1, the rule gives 5.4366, with a calculator.

dy/dx at x = 1: 5.4366

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

A calculator gives about 5.44.

Very close to 5.4366.

Right!

y(1.001) − y(1)0.001 ≈ 5.44

Let's look back.

First: the inside.

Then: the outside.

Then: the inside's slope.

Then: multiply.

Last: check.

Answer