Derivative of sin^-1(2x^3) Step by Step

Step by step

The question

Question: find the derivative of f(x) = sin⁻¹(2x³).

f(x) = sin⁻¹(2x³)

1 The rule

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

sin⁻¹ u → 1√1 − u²

2 The inside

There is something inside: 2x³.

Call it u.

u = 2x³

3 The outside

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

dydu = 1√1 − u²

4 The inside's slope

The inside's slope: the power rule on 2x³ gives 6x².

dudx = 6x²

5 Multiply

The chain rule: multiply the two, and put 2x³ back in place of u.

f′(x) = 1√1 − (2x³)² × 6x²

Careful!

Do not stop at the outside.

Multiply by the inside's slope, 6x².

Careful: not 1√1 − (2x³)² ✗

Square the inside: (2x³)² is 4x⁶.

(2x³)² = 4x⁶

Tidy up: f′(x) = 6x² over √(1 − 4x⁶).

= 6x²√1 − 4x⁶

6 Check

Let's check: the derivative is the slope.

At x = 0.5, our answer gives 1.5492, with a calculator in radians.

f′(0.5) = 1.5492

Now the slope between two very close points: x = 0.5 and x = 0.501.

A calculator gives about 1.55.

Very close to 1.5492.

Right!

f(0.501) − f(0.5)0.001 ≈ 1.55

Let's look back.

First: the rule.

Then: the inside.

Then: the outside.

Then: the inside's slope.

Then: multiply.

Last: check.

Answer