Derivative of 5cos^-1(x) Step by Step

Step by step

Question: find the derivative of f(x) = 5cos⁻¹ x.

f(x) = 5cos⁻¹ x

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!

The little −1 means the inverse function, the angle.

It does not mean 1 over cos x.

Careful: cos⁻¹ x is not 1cos x

The 5 in front just waits, and multiplies.

f′(x) = 5 × −1√1 − x²

So f′(x) = −5 over √(1 − x²).

= −5√1 − x²

Let's check: the derivative is the slope.

At x = 0.5, our answer gives −5.7735, with a calculator in radians.

f′(0.5) = −5.7735

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

A calculator gives about −5.78.

Very close to −5.7735.

Right!

f(0.501) − f(0.5)0.001 ≈ −5.78

Let's look back.

First: the rule.

Then: use the rule.

Last: check.

Answer

= −5√1 − x²