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

Step by step

The question

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

f(x) = cos⁻¹(x²)

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: x².

Call it u.

u = x²

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 x² gives 2x.

dudx = 2x

5 Multiply

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

f′(x) = −1√1 − (x²)² × 2x

Careful!

Do not stop at the outside.

Multiply by the inside's slope, 2x.

Careful: not −1√1 − (x²)² ✗

Square the inside: (x²)² is x⁴.

(x²)² = x⁴

Tidy up: f′(x) = −2x over √(1 − x⁴).

= −2x√1 − x⁴

6 Check

Let's check: the derivative is the slope.

At x = 0.5, our answer gives −1.0328, with a calculator in radians.

f′(0.5) = −1.0328

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

A calculator gives about −1.03.

Very close to −1.0328.

Right!

f(0.501) − f(0.5)0.001 ≈ −1.03

Let's look back.

First: the rule.

Then: the inside.

Then: the outside.

Then: the inside's slope.

Then: multiply.

Last: check.

Answer