Step by step
The question
Question: find the derivative of f(x) = sin⁻¹ x.
f(x) = sin⁻¹ x
1 The rule
y = sin⁻¹ x means: sin y = x.
y is the angle whose sine is x.
y = sin⁻¹ x means sin y = x
Take the derivative of both sides: cos y times y′ is 1.
So y′ is 1 over cos y.
And cos y is the square root of 1 minus sin² y, which is 1 minus x².
cos y × y′ = 1
y′ = 1cos y = 1√1 − x²
The rule: the derivative of sin⁻¹ x is 1 over √(1 − x²).
sin⁻¹ x → 1√1 − x²
Careful!
The little −1 means the inverse function, the angle.
It does not mean 1 over sin x.
Careful: sin⁻¹ x is not 1sin x
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