Derivative of 3sin^-1(x) Step by Step

Step by step

Question: find the derivative of f(x) = 3sin⁻¹ x.

f(x) = 3sin⁻¹ x

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

The 3 in front just waits, and multiplies.

f′(x) = 3 × 1√1 − x²

So f′(x) = 3 over √(1 − x²).

= 3√1 − x²

Let's check: the derivative is the slope.

At x = 0.5, our answer gives 3.4641, with a calculator in radians.

f′(0.5) = 3.4641

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

A calculator gives about 3.47.

Very close to 3.4641.

Right!

f(0.501) − f(0.5)0.001 ≈ 3.47

Let's look back.

First: the rule.

Then: use the rule.

Last: check.

Answer

= 3√1 − x²