Derivative of acos(x) - Proof and Explanation

To find the derivative of arccos(x), we start by letting:

y=arccos(x)

This means that y is the angle whose cosine is x. In other words, we have:

x=cos(y)

Step 1: Differentiate both sides

We differentiate both sides of this equation with respect to x. The left side simply becomes 1. For the right side, using the chain rule:

ddx(cos(y))=sin(y)dydx

So, the equation becomes:

1=sin(y)dydx

Step 2: Solve for dydx

Rearranging the equation to solve for dydx:

dydx=1sin(y)

Step 3: Express sin(y) in terms of x

Using the Pythagorean identity:

sin2(y)+cos2(y)=1

We can express sin(y) as:

sin(y)=1cos2(y)=1x2

Step 4: Substitute sin(y) back into the derivative

Substituting this back into our expression for dydx:

dydx=11x2

Thus, the derivative of arccos(x) is:

11x2

Q.E.D.