A Comprehensive Guide to Finding the Derivative of the Function arcsec(x)

d/dx arcsec(x)

To find the derivative of the function arcsec(x) with respect to x, we can start by expressing the function in terms of inverse trigonometric functions and then differentiate using the chain rule

To find the derivative of the function arcsec(x) with respect to x, we can start by expressing the function in terms of inverse trigonometric functions and then differentiate using the chain rule.

The function arcsec(x) is the inverse of the secant function and can be defined as follows:

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

Now, let’s differentiate arcsec(x) with respect to x:

Using the chain rule, we have:

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

To simplify the problem, we can express sec^(-1)(x) in terms of the cosine function. Recall that the reciprocal identity for secant is:

sec^(-1)(x) = cos^(-1)(1/x)

Now, we need to differentiate cos^(-1)(1/x):

d/dx [cos^(-1)(1/x)]

Let’s denote u = 1/x. Then u is a function of x. We can rewrite the expression as:

cos^(-1)(u)

Now, we’ll differentiate cos^(-1)(u) with respect to u:

d/du [cos^(-1)(u)] = -1/sqrt(1 – u^2)

Finally, we substitute back u = 1/x:

d/dx [arcsec(x)] = -1/sqrt(1 – (1/x)^2)

Simplifying it further, we have:

d/dx [arcsec(x)] = -1/sqrt(1 – 1/x^2)

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

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