How to Find the Derivative of the Inverse Secant Function: Step-by-Step Guide

d/dx arcsec(x)

To find the derivative of the inverse secant function, arcsec(x), with respect to x, we first need to use the definition of the inverse secant function and express it in terms of another trigonometric function

To find the derivative of the inverse secant function, arcsec(x), with respect to x, we first need to use the definition of the inverse secant function and express it in terms of another trigonometric function.

The inverse secant function is defined as the inverse of the secant function:

arcsec(x) = sec^(-1)(x)

Now, let’s start by finding the derivative of sec(x), since we will need it to find the derivative of arcsec(x).

The derivative of sec(x) can be derived using the quotient rule:

d/dx sec(x) = d/dx (1/cos(x)) = -1/cos(x) * -sin(x)/cos^2(x) = sin(x)/cos^2(x) = tan(x)sec(x)

So, we have:

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

Now, using the definition of the inverse function and the chain rule, we can find the derivative of arcsec(x):

d/dx arcsec(x) = d/dx sec^(-1)(x)

Let y = sec^(-1)(x), then x = sec(y)

Differentiating both sides with respect to x:

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

Dividing both sides by sec(y) * tan(y):

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

Now, we can substitute x = sec(y) back into the equation:

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

Finally, using a trigonometric identity, we can simplify the expression further. Recall that tan(arcsec(x)) is equal to √(x^2 – 1). Therefore, we have:

dy/dx = 1 / (x * √(x^2 – 1))

So, the derivative of arcsec(x) with respect to x is 1 / (x * √(x^2 – 1)).

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