Derivative of Arctan u

We understand the derivative of arctan(u). It is commonly represented as d/du (arctan(u)) or (arctan(u))', and its value is 1/(1 + u²). The function arctan(u) has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Arctangent Function: (arctan(u) is the inverse of tan(u)). Chain Rule: Rule for differentiating composite functions like arctan(u). Reciprocal Function: A function that returns the reciprocal of its input, used in deriving the formula.

The derivative of arctan(u) can be denoted as d/du (arctan(u)) or (arctan(u))'. The formula we use to differentiate arctan(u) is: d/du (arctan(u)) = 1/(1 + u²) This formula applies to all u where the function is defined.

We can derive the derivative of arctan(u) 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 We will now demonstrate that differentiating arctan(u) results in 1/(1 + u²) using the above-mentioned methods: By First Principle The derivative of arctan(u) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of arctan(u) using the first principle, we will consider f(u) = arctan(u). Its derivative can be expressed as the following limit. f'(u) = limₕ→₀ [f(u + h) - f(u)] / h Given that f(u) = arctan(u), we write f(u + h) = arctan(u + h). Substituting these into the limit equation, we use trigonometric identities to simplify: f'(u) = limₕ→₀ [arctan(u + h) - arctan(u)] / h = limₕ→₀ [tan(arctan(u + h)) - tan(arctan(u))] / [h(1 + u²)] Using the identity tan(arctan(x)) = x, we simplify further to show: f'(u) = 1/(1 + u²) Hence, proved. Using Chain Rule To prove the differentiation of arctan(u) using the chain rule, We use the identity that relates arctan(u) to its derivative: Let y = arctan(u) ⇒ tan(y) = u Differentiating both sides with respect to u, using implicit differentiation: sec²(y) dy/du = 1 Using the identity 1 + tan²(y) = sec²(y), substitute tan(y) = u: (1 + u²) dy/du = 1 Thus, dy/du = 1/(1 + u²) 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(u). For the first derivative of a function, we write f′(u), 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′′(u). Similarly, the third derivative, f′′′(u), is the result of the second derivative, and this pattern continues. For the nth Derivative of arctan(u), we generally use f⁽ⁿ⁾(u) for the nth derivative of a function f(u), which tells us the change in the rate of change, continuing for higher-order derivatives.

When u approaches ±∞, the derivative approaches 0 because the slope of arctan(u) flattens out. When u = 0, the derivative of arctan(u) = 1/(1 + 0²), which is 1.

Derivative: The derivative of a function indicates how the given function changes in response to a slight change in its input. Arctangent Function: The arctangent function is the inverse of the tangent function and is written as arctan(u). Chain Rule: A rule for differentiating compositions of functions. First Derivative: It is the initial result of a function, which gives us the rate of change of a specific function. Implicit Differentiation: A method used to find derivatives of functions not explicitly solved for one variable in terms of another.