Derivative of Cos Inverse

We now understand the derivative of cos⁻¹(x). It is commonly represented as d/dx (cos⁻¹(x)) or (cos⁻¹(x))', and its value is -1/√(1-x²). The function cos⁻¹(x) has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Inverse Cosine Function: (cos⁻¹(x)) is the inverse of the cosine function. Chain Rule: Rule for differentiating cos⁻¹(x). Square Root Function: √(1-x²) appears in the derivative.

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

We can derive the derivative of cos⁻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 Trigonometric Identities We will now demonstrate that the differentiation of cos⁻¹(x) results in -1/√(1-x²) using the above-mentioned methods: By First Principle The derivative of cos⁻¹(x) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of cos⁻¹(x) using the first principle, we will consider f(x) = cos⁻¹(x). Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = cos⁻¹(x), we write f(x + h) = cos⁻¹(x + h). Substituting these into equation (1), f'(x) = limₕ→₀ [cos⁻¹(x + h) - cos⁻¹(x)] / h Using the identity cos(cos⁻¹(x)) = x, and differentiating implicitly, we find: f'(x) = -1/√(1-x²) Hence, proved. Using Chain Rule To prove the differentiation of cos⁻¹(x) using the chain rule, We use the formula: cos(cos⁻¹(x)) = x Differentiating both sides with respect to x, we get: -sin(cos⁻¹(x)) * d/dx(cos⁻¹(x)) = 1 Solving for d/dx(cos⁻¹(x)), we find: d/dx(cos⁻¹(x)) = -1/√(1-x²) Using Trigonometric Identities We can also use trigonometric identities to prove the derivative of cos⁻¹(x). If y = cos⁻¹(x), then x = cos(y). Differentiating both sides with respect to x: 1 = -sin(y) * dy/dx So, dy/dx = -1/sin(y) Since sin(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 cos⁻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 cos⁻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 = 1 or x = -1, the derivative is undefined because cos⁻¹(x) is not differentiable at these points. When x = 0, the derivative of cos⁻¹(x) = -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 x. Inverse Cosine Function: The inverse cosine function, denoted as cos⁻¹(x), is the angle whose cosine is x. Chain Rule: A rule in calculus used to differentiate compositions of functions. Square Root Function: The function √(1-x²) appears in the derivative of inverse trigonometric functions. Asymptote: A line that a curve approaches as it heads towards infinity.