Derivative of Tan⁻¹x

We now understand the derivative of tan⁻¹x. It is commonly represented as d/dx (tan⁻¹x) or (tan⁻¹x)', and its value is 1/(1 + x²). The function tan⁻¹x has a clearly defined derivative, indicating it is differentiable for all real numbers.

The key concepts are mentioned below: Inverse Tangent Function: (tan⁻¹(x) is the inverse of tan(x)). Chain Rule: A rule for differentiating composite functions. Secant Function: sec(x) = 1/cos(x).

The derivative of tan⁻¹x can be denoted as d/dx (tan⁻¹x) or (tan⁻¹x)'. The formula we use to differentiate tan⁻¹x is: d/dx (tan⁻¹x) = 1/(1 + x²)

The formula applies to all real x.

We can derive the derivative of tan⁻¹x using proofs. To show this, we will use implicit differentiation and trigonometric identities.

There are several methods to prove this, such as:

• Using Implicit Differentiation
   
• Using Trigonometric Identities

We will now demonstrate that the differentiation of tan⁻¹x results in 1/(1 + x²) using these methods:

Using Implicit Differentiation

Consider y = tan⁻¹x. This implies x = tan(y). Differentiating both sides with respect to x gives: 1 = sec²(y) · dy/dx Recall that sec²(y) = 1 + tan²(y). Thus, 1 = (1 + x²) dy/dx (since tan(y) = x) Therefore, dy/dx = 1/(1 + x²).

Using Trigonometric Identities

We know the identity tan²(y) + 1 = sec²(y). Therefore, using implicit differentiation, we substitute tan(y) = x, resulting in: 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 bit complex. 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 help analyze functions like tan⁻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 tan⁻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 = 0, the derivative of tan⁻¹x = 1/(1 + 0²), which is 1.

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

• Inverse Tangent Function: The function tan⁻¹(x) is the inverse of the tangent function and is used to find the angle whose tangent is x.

• Implicit Differentiation: A technique used to differentiate equations where the dependent variable is not isolated on one side of the equation.

• Chain Rule: A rule used to differentiate composite functions, where one function is nested inside another.

• Quotient Rule: A rule used to differentiate functions of the form u/v, where both u and v are differentiable functions of x.