Step by step
The question
Question: find the derivative of f(x) = cos⁻¹ x.
f(x) = cos⁻¹ x
1 The rule
y = cos⁻¹ x means: cos y = x.
y is the angle whose cosine is x.
y = cos⁻¹ x means cos y = x
Take the derivative of both sides: minus sin y, times y′, is 1.
So y′ is minus 1 over sin y.
And sin y is the square root of 1 minus x².
−sin y × y′ = 1
y′ = −1sin y = −1√1 − x²
The rule: the derivative of cos⁻¹ x is −1 over √(1 − x²).
Mind the minus!
cos⁻¹ x → −1√1 − x²
Careful!
cos⁻¹ has a minus in its derivative.
Do not lose it.
Careful: not cos⁻¹ x → 1√1 − x² ✗
2 Use the rule
So f′(x) = −1 over √(1 − x²).
f′(x) = −1√1 − x²
3 Check
Let's check: the derivative is the slope.
At x = 0.5, our answer gives −1.1547, with a calculator in radians.
f′(0.5) = −1.1547
Now the slope between two very close points: x = 0.5 and x = 0.501.
A calculator gives about −1.16.
Very close to −1.1547.
Right!
f(0.501) − f(0.5)0.001 ≈ −1.16
Let's look back.
First: the rule.
Then: use the rule.
Last: check.
Answer
f′(x) = −1√1 − x²