y = cos^-1(3x), find dy/dx Step by Step

Step by step

The question

Question: find dy/dx for y = cos⁻¹(3x).

y = cos⁻¹(3x)

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: 3x.

Call it u.

u = 3x

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 3x gives 3.

dudx = 3

5 Multiply

The chain rule: multiply the two, and put 3x back in place of u.

dydx = −1√1 − (3x)² × 3

Careful!

Do not stop at the outside.

Multiply by the inside's slope, 3.

Careful: not −1√1 − (3x)² ✗

Square the inside: (3x)² is 9x².

(3x)² = 9x²

Tidy up: dy/dx = −3 over √(1 − 9x²).

= −3√1 − 9x²

6 Check

Let's check: the derivative is the slope.

At x = 0.2, our answer gives −3.75, with a calculator in radians.

dy/dx at x = 0.2: −3.75

Now the slope between two very close points: x = 0.2 and x = 0.201.

A calculator gives about −3.76.

Very close to −3.75.

Right!

y(0.201) − y(0.2)0.001 ≈ −3.76

Let's look back.

First: the rule.

Then: the inside.

Then: the outside.

Then: the inside's slope.

Then: multiply.

Last: check.

Answer