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

Step by step

The question

Question: find dy/dx for y = cos(2x).

y = cos(2x)

1 The inside

There is something inside: 2x.

Call it u.

u = 2x

Then y is just cos u.

y = cos u

2 The outside

The derivative of cos x is −sin x.

Mind the minus!

cos x → −sin x

First the outside.

The rule on cos u gives −sin u.

dydu = −sin u

3 The inside's slope

Now the inside.

The power rule on 2x gives 2.

dudx = 2

4 Multiply

The chain rule: multiply the two.

dy/du times du/dx.

dydx = dydu × dudx

Put the inside back in place of u.

= −sin(2x) × 2

Careful!

Do not stop here!

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

Careful: not −sin(2x) ✗

−1 × 2 = −2

Write the numbers in front.

So dy/dx = −2 sin(2x).

= −2 sin(2x)

5 Check

Let's check: the derivative is the slope.

At x = 1, the rule gives −1.8186, with a calculator in radians.

dy/dx at x = 1: −1.8186

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

A calculator gives about −1.82.

Very close to −1.8186.

Right!

y(1.001) − y(1)0.001 ≈ −1.82

Let's look back.

First: the inside.

Then: the outside.

Then: the inside's slope.

Then: multiply.

Last: check.

Answer