Derivative of arcsec(x) and Chain Rule Explanation

(d/dx) arcsec(x)

To find the derivative of the function arcsec(x), which represents the inverse secant function, we can use the chain rule

To find the derivative of the function arcsec(x), which represents the inverse secant function, we can use the chain rule.

Let’s first denote y = arcsec(x). This means x = sec(y), where y lies in the domain of arcsec(x).

Now, we can differentiate both sides of the equation with respect to x:

d/dx(x) = d/dx(sec(y))

Using the chain rule, we can break down the derivative of sec(y) as follows:

1 = sec(y) * tan(y) * dy/dx

We know that sec(y) = x, so we can substitute it in the equation:

1 = x * tan(y) * dy/dx

Now, to solve for dy/dx, we can move some terms around:

dy/dx = 1 / (x * tan(y))

Recall that tan(y) can be expressed in terms of x because we know x = sec(y):

dy/dx = 1 / (x * tan(y)) = 1 / (x * sqrt((sec^2(y) – 1))

Using the Pythagorean identity sec^2(y) – 1 = tan^2(y):

dy/dx = 1 / (x * sqrt((tan^2(y)) = 1 / (x * sqrt((x^2 – 1))

Therefore, the derivative of arcsec(x) is given by:

(d/dx) arcsec(x) = dy/dx = 1 / (x * sqrt((x^2 – 1))

This is the detailed answer for the derivative of arcsec(x).

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