Derivative of cos^-1(2x) Step by Step

Step by step

The question

Question: find the derivative of f(x) = cos⁻¹(2x).

f(x) = cos⁻¹(2x)

1 The rule

The rule: the derivative of cos⁻¹ u is −1 over √(1 − u²).

Mind the minus!

cos⁻¹ u → −1√1 − u²

2 The inside

There is something inside: 2x.

Call it u.

u = 2x

3 The outside

First the outside: the rule on cos⁻¹ u.

dydu = −1√1 − u²

4 The inside's slope

The inside's slope: the power rule on 2x gives 2.

dudx = 2

5 Multiply

The chain rule: multiply the two, and put 2x back in place of u.

f′(x) = −1√1 − (2x)² × 2

Careful!

Do not stop at the outside.

Multiply by the inside's slope, 2.

Careful: not −1√1 − (2x)² ✗

Square the inside: (2x)² is 4x².

(2x)² = 4x²

Tidy up: f′(x) = −2 over √(1 − 4x²).

= −2√1 − 4x²

6 Check

Let's check: the derivative is the slope.

At x = 0.4, our answer gives −3.3333, with a calculator in radians.

f′(0.4) = −3.3333

Now the slope between two very close points: x = 0.4 and x = 0.401.

A calculator gives about −3.34.

Very close to −3.3333.

Right!

f(0.401) − f(0.4)0.001 ≈ −3.34

Let's look back.

First: the rule.

Then: the inside.

Then: the outside.

Then: the inside's slope.

Then: multiply.

Last: check.

Answer