Derivative of Arccot x

We now understand the derivative of arccot x. It is commonly represented as d/dx (arccot x) or (arccot x)', and its value is -1/(1 + x²). The function arccot x has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Arccotangent Function: The inverse of the cotangent function. Chain Rule: Useful for differentiating composite functions like arccot(x). Pythagorean Identity: Involves the relationship between sine, cosine, and tangent, essential for trigonometric derivatives.

The derivative of arccot x can be denoted as d/dx (arccot x) or (arccot x)'. The formula we use to differentiate arccot x is: d/dx (arccot x) = -1/(1 + x²) The formula applies to all x where x is defined for arccot.

We can derive the derivative of arccot x using proofs. To show this, we will use trigonometric identities along with the rules of differentiation. There are several methods we use to prove this, such as: By First Principle Using Chain Rule Using Implicit Differentiation We will now demonstrate that the differentiation of arccot x results in -1/(1 + x²) using the above-mentioned methods: By First Principle The derivative of arccot x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of arccot x using the first principle, we will consider f(x) = arccot x. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h Given that f(x) = arccot x, we write f(x + h) = arccot (x + h). Substituting these into the equation, f'(x) = limₕ→₀ [arccot(x + h) - arccot x] / h Using trigonometric identities, we know that arccot(x) is the angle whose cotangent is x, so we use implicit differentiation. Using Chain Rule To prove the differentiation of arccot x using the chain rule, We use the identity: arccot(x) = π/2 - arctan(x) Differentiate implicitly: d/dx [π/2 - arctan(x)] = -d/dx [arctan(x)] d/dx [arctan(x)] = 1/(1 + x²) Thus, d/dx [arccot(x)] = -1/(1 + x²) Using Implicit Differentiation We will prove the derivative of arccot x using implicit differentiation. The step-by-step process is demonstrated below: Consider y = arccot(x), then cot(y) = x. Differentiating both sides with respect to x gives: -csc²(y) dy/dx = 1 Thus, dy/dx = -1/(csc²(y)) Since csc²(y) = 1 + x², we have dy/dx = -1/(1 + x²)

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 arccot(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 arccot(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 0, the derivative of arccot x = -1/(1 + 0²), which is -1. When x approaches infinity, the derivative approaches 0, as arccot(x) tends to 0.

Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Arccotangent Function: The inverse of the cotangent function, symbolized as arccot x. Chain Rule: A rule used to differentiate composite functions, essential in finding the derivative of inverse functions. Implicit Differentiation: A method to differentiate equations not explicitly solved for one variable in terms of another. Pythagorean Identity: A fundamental relation between trigonometric functions, used in many derivative proofs.