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

Step by step

The question

Question: find the derivative of f(x) = 2sin⁻¹(x²).

1 The rule

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

2 The inside

There is something inside: x².

Call it u.

3 The outside

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

The 2 waits.

4 The inside's slope

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

5 Multiply

The chain rule: multiply the two, and put x² back in place of u.

Careful!

Do not stop at the outside.

Multiply by the inside's slope, 2x.

Square the inside: (x²)² is x⁴.

Tidy up: f′(x) = 4x over √(1 − x⁴).

6 Check

Let's check: the derivative is the slope.

At x = 0.5, our answer gives 2.0656, with a calculator in radians.

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

A calculator gives about 2.07.

Very close to 2.0656.

Right!

Let's look back.

First: the rule.

Then: the inside.

Then: the outside.

Then: the inside's slope.

Then: multiply.

Last: check.