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

Step by step

The question

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

f(x) = sin⁻¹(2x − 1)

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: 2x − 1.

Call it u.

u = 2x − 1

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 2x − 1 gives 2.

dudx = 2

5 Multiply

The chain rule: multiply the two, and put 2x − 1 back in place of u.

f′(x) = 1√1 − (2x − 1)² × 2

Careful!

Do not stop at the outside.

Multiply by the inside's slope, 2.

Careful: not 1√1 − (2x − 1)² ✗

Tidy up: f′(x) = 2 over (√(1 − (2x − 1)²)).

= 2√1 − (2x − 1)²

6 Check

Let's check: the derivative is the slope.

At x = 0.5, our answer gives 2, with a calculator in radians.

f′(0.5) = 2

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

A calculator gives about 2.

Very close to 2.

Right!

f(0.501) − f(0.5)0.001 ≈ 2

Let's look back.

First: the rule.

Then: the inside.

Then: the outside.

Then: the inside's slope.

Then: multiply.

Last: check.

Answer