Derivative of asin(x) - Proof and Explanation

To find the derivative of $\arcsin (x)$ , we start by letting:

y = \arcsin (x)

This implies that:

\sin (y) = x

Step 1: Differentiate both sides

Differentiating $\sin (y) = x$ with respect to $x$ , we apply implicit differentiation:

\cos (y) \frac{d y}{d x} = 1

Here, $\cos (y)$ is the derivative of $\sin (y)$ with respect to $y$ , and $\frac{d y}{d x}$ is the derivative of $y$ with respect to $x$ .

Step 2: Solve for $\frac{d y}{d x}$

Rearrange the equation to solve for the derivative:

\frac{d y}{d x} = \frac{1}{\cos (y)}

Step 3: Express $\cos (y)$ in terms of $x$

We use the Pythagorean identity:

\cos^{2} (y) = 1 - \sin^{2} (y)

\cos (y) = \sqrt{1 - \sin^{2} (y)}

Since $\sin (y) = x$ , we substitute:

\cos (y) = \sqrt{1 - x^{2}}

Step 4: Final derivative

Substituting $\cos (y)$ back into the expression for $\frac{d y}{d x}$ :

\frac{d y}{d x} = \frac{1}{\sqrt{1 - x^{2}}}

Therefore, the derivative of $\arcsin (x)$ is:

\frac{1}{\sqrt{1 - x^{2}}}

Q.E.D.