Derivative of ln(sin x)

We now understand the derivative of ln(sin x). It is commonly represented as d/dx (ln(sin x)) or (ln(sin x))', and its value is cot x. The function ln(sin x) has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Logarithmic Function: ln(x) is the natural logarithm function.

Chain Rule: Rule for differentiating ln(sin x) (since it involves a composition of functions).

Cotangent Function: cot(x) = 1/tan(x).

The derivative of ln(sin x) can be denoted as d/dx (ln(sin x)) or (ln(sin x))'. The formula we use to differentiate ln(sin x) is: d/dx (ln(sin x)) = cot x (or) (ln(sin x))' = cot x

The formula applies to all x where sin(x) > 0

We can derive the derivative of ln(sin x) using proofs. To show this, we will use the 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 Product Rule

We will now demonstrate that the differentiation of ln(sin x) results in cot x using the above-mentioned methods:

By First Principle

The derivative of ln(sin x) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of ln(sin x) using the first principle, we will consider f(x) = ln(sin 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) = ln(sin x), we write f(x + h) = ln(sin(x + h)). Substituting these into equation (1), f'(x) = limₕ→₀ [ln(sin(x + h)) - ln(sin x)] / h = limₕ→₀ ln([sin(x + h) / sin x]) / h Using the logarithmic identity ln(a) - ln(b) = ln(a/b), f'(x) = limₕ→₀ [ln(1 + (sin(x + h) - sin x)/sin x)] / h Using the limit definition, f'(x) = limₕ→₀ [(sin(x + h) - sin x)/(h sin x)] Using the identity sin(A + h) - sin A = 2 cos((2A + h)/2) sin(h/2), f'(x) = limₕ→₀ [cos(x + h/2) sin(h/2)/sin x] / (h/2) Using limit formulas, limₕ→₀ sin(h/2)/(h/2) = 1. f'(x) = cos x/sin x = cot x.

Using Chain Rule

To prove the differentiation of ln(sin x) using the chain rule, We use the formula: Let u = sin x, then ln(u) Differentiate using the chain rule: d(ln(u))/dx = 1/u * du/dx So, d(ln(sin x))/dx = 1/sin x * cos x = cot x.

Using Product Rule

We will now prove the derivative of ln(sin x) using the product rule. Let y = ln(sin x) = ln u, where u = sin x. Differentiate using the chain rule: dy/dx = 1/u * du/dx = 1/sin x * cos x = cot x.

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 ln(sin 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 ln(sin 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 ln(sin x) is undefined for sin x = 0.

When x is π/2, the derivative of ln(sin x) = cot(π/2), which is 0.

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

• Logarithmic Function: A function that uses logarithms, often denoted by ln(x).

• Chain Rule: A rule for differentiating compositions of functions, used in finding the derivative of ln(sin x).

• Cotangent Function: A trigonometric function that is the reciprocal of the tangent function, written as cot x.

• Undefined Points: Points at which a function is not defined, such as where sin(x) = 0 for ln(sin x).