Derivative of arctan(1/x)

We now understand the derivative of arctan(1/x). It is commonly represented as d/dx (arctan(1/x)) or (arctan(1/x))', and its value is -1/(x² + 1).

The function arctan(1/x) has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below: Inverse Tangent Function: arctan(y) = x implies tan(x) = y.

Chain Rule: Rule for differentiating composite functions like arctan(1/x). Derivative of arctan(x): d/dx (arctan(x)) = 1/(1 + x²).

The derivative of arctan(1/x) can be denoted as d/dx (arctan(1/x)) or (arctan(1/x))'.

The formula we use to differentiate arctan(1/x) is: d/dx (arctan(1/x)) = -1/(x² + 1)

The formula applies to all x where x ≠ 0.

We can derive the derivative of arctan(1/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 arctan(1/x) results in -1/(x² + 1)

using the above-mentioned methods: Using Chain Rule To prove the differentiation of arctan(1/x) using the chain rule, We use the formula: u = 1/x and f(u) = arctan(u)

Thus, y = arctan(1/x) = arctan(u) By chain rule: dy/dx = dy/du * du/dx We know dy/du = 1/(1 + u²) and du/dx = -1/x² Substitute u = 1/x to get: dy/dx = 1/(1 + (1/x)²) * (-1/x²) = -1/(x² + 1)

Thus, the derivative is -1/(x² + 1). Using Implicit Differentiation Consider y = arctan(1/x).

Then, tan(y) = 1/x. Differentiating both sides with respect to x, using implicit differentiation: sec²(y) * dy/dx = -1/x² Since sec²(y) = 1 + tan²(y), substitute tan(y) = 1/x: (1 + (1/x)²) * dy/dx = -1/x² dy/dx = -1/(x² + 1)

Hence, proved.

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 arctan(1/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 arctan(1/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 is undefined because arctan(1/x) has a vertical asymptote there. When x is 1, the derivative of arctan(1/x) = -1/(1² + 1) = -1/2.

• Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.

• Inverse Tangent Function: The inverse of the tangent function, represented as arctan(x), gives the angle whose tangent is x.

• Chain Rule: A fundamental rule for differentiating composite functions, allowing us to find the derivative of functions like arctan(1/x).

• Quotient Rule: A method used to differentiate functions that are divided by one another, such as arctan(1/x)/x.

• Asymptote: A line that a function approaches but never crosses, relevant for points where functions like arctan(1/x) become undefined.