Derivative of Arcsec

We now understand the derivative of arcsec(x). It is commonly represented as d/dx (arcsec x) or (arcsec x)', and its value is 1/(|x|√(x²-1)). The function arcsec x has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Arcsecant Function: arcsec(x) is the inverse function of sec(x). Inverse Function Rule: Rule for differentiating arcsec(x) using its relationship with sec(x). Absolute Value: |x| is used in the formula to ensure the result is valid for both positive and negative x.

The derivative of arcsec(x) can be denoted as d/dx (arcsec x) or (arcsec x)'. The formula we use to differentiate arcsec x is: d/dx (arcsec x) = 1/(|x|√(x²-1)) The formula applies to all x where |x| > 1.

We can derive the derivative of arcsec x using proofs. To show this, we will use the trigonometric identities along with the rules of differentiation. There are several methods we use to prove this, such as: By First Principle Using Inverse Function Rule Using Implicit Differentiation We will now demonstrate that the differentiation of arcsec x results in 1/(|x|√(x²-1)) using the above-mentioned methods: Using Inverse Function Rule The derivative of arcsec x can be proved using the inverse function rule. If y = arcsec x, then x = sec y. Differentiating both sides with respect to x, we have: d/dx (x) = d/dx (sec y) 1 = sec y tan y (dy/dx) We know from trigonometric identities that sec² y - 1 = tan² y, hence tan y = √(sec² y - 1). Substituting sec y = x: 1 = x √(x² - 1) (dy/dx) dy/dx = 1/(x √(x² - 1)) Since y = arcsec x, the derivative is: d/dx (arcsec x) = 1/(|x|√(x²-1)).

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky. To understand them better, think of a car where the speed changes (first derivative) and the rate at which the speed changes (second derivative) also changes. Higher-order derivatives make it easier to understand functions like arcsec(x). For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point. The second derivative is derived from the first derivative, which is denoted using f′′(x). Similarly, the third derivative, f′′′(x), is the result of the second derivative and this pattern continues. For the nth Derivative of arcsec(x), we generally use fⁿ(x) for the nth derivative of a function f(x), which tells us the change in the rate of change (continuing for higher-order derivatives).

When x is 1 or -1, the derivative is undefined because arcsec(x) is not defined there. When x is √2, the derivative of arcsec x = 1/(|√2|√((√2)²-1)), which simplifies to 1/√2.

Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Arcsecant Function: The arcsecant function is the inverse of the secant function and is written as arcsec x. Inverse Function Rule: A method used to find the derivative of inverse functions. Chain Rule: A rule for finding the derivative of a composition of functions. Absolute Value: The non-negative value of a number without regard to its sign, denoted as |x|.