Master the Chain Rule: Derivative of the Arcsine Function Explained

d/dx arcsin(x)

To find the derivative of the arcsine function, we can use the chain rule.

The derivative of the arcsine function, denoted as d/dx of arcsin(x), can be written as:

d/dx arcsin(x) = 1 / (√(1 – x^2))

To prove this, let’s start by using the definition of the arcsine function.

The arcsine function can be defined as the inverse of the sine function. This means that for any value of x, if y = arcsin(x), then sin(y) = x.

In other words, x = sin(y).

We can rewrite this equation as x = sin(arcsin(x)).

Now, let’s differentiate both sides of this equation with respect to x using the chain rule.

d/dx (x) = d/dx (sin(arcsin(x)).

The derivative of x with respect to x is simply 1.

Therefore, 1 = d/dx (sin(arcsin(x)).

Now, to simplify further, we need to compute the derivative of sin(arcsin(x)).

Using the chain rule, we have:

d/dx (sin(arcsin(x))) = cos(arcsin(x)) * d/dx (arcsin(x)).

Now, we need to find the derivative of arcsin(x) with respect to x.

To do this, we can express arcsin(x) as a function of y, where y = arcsin(x), and rewrite the equation as x = sin(y).

Differentiating both sides of this equation with respect to x using the chain rule, we have:

1 = cos(y) * dy/dx.

Since y = arcsin(x), we can rewrite this equation as:

1 = cos(arcsin(x)) * dy/dx.

Simplifying this equation, we have:

1 = √(1 – x^2) * dy/dx.

Dividing both sides of the equation by √(1 – x^2), we get:

dy/dx = 1 / √(1 – x^2).

Substituting this result back into our previous equation, we have:

1 = cos(arcsin(x)) * (1 / √(1 – x^2)).

Since cos(arcsin(x)) is equal to √(1 – x^2) (from the definition of the cosine function), our equation becomes:

1 = √(1 – x^2) * (1 / √(1 – x^2)).

Simplifying this equation, we have:

1 = 1.

This equation holds true for all values of x, so our derivative is:

d/dx arcsin(x) = 1 / √(1 – x^2).

Therefore, the derivative of the arcsine function is given by 1 divided by the square root of 1 minus x squared.