Derivative of ln(x/2)

We now understand the derivative of ln(x/2). It is commonly represented as d/dx (ln(x/2)) or (ln(x/2))', and its value is 1/x. The function ln(x/2) has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Logarithmic Function: ln(x/2) = ln(x) - ln(2).

Quotient Rule: Rule for differentiating ln(x/2) (since it consists of ln(x) - ln(2)).

Natural Logarithm: ln(x) is the logarithm to the base e, a mathematical constant.

The derivative of ln(x/2) can be denoted as d/dx (ln(x/2)) or (ln(x/2))'. The formula we use to differentiate ln(x/2) is: d/dx (ln(x/2)) = 1/x

The formula applies to all x where x > 0.

We can derive the derivative of ln(x/2) using proofs. To show this, we will use logarithmic 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 Properties of Logarithms

We will now demonstrate that the differentiation of ln(x/2) results in 1/x using the above-mentioned methods:

By First Principle

The derivative of ln(x/2) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of ln(x/2) using the first principle, we will consider f(x) = ln(x/2). Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h Given that f(x) = ln(x/2), we write f(x + h) = ln((x + h)/2). Substituting these into the equation, f'(x) = limₕ→₀ [ln((x + h)/2) - ln(x/2)] / h = limₕ→₀ [ln((x + h)/x)] / h = limₕ→₀ ln(1 + h/x)/h = (1/x) limₕ→₀ ln(1 + h/x)/(h/x) Using limit formulas, limₕ→₀ ln(1 + u)/u = 1 as u → 0. f'(x) = 1/x Hence, proved.

Using Chain Rule

To prove the differentiation of ln(x/2) using the chain rule, We use the formula: ln(x/2) = ln(x) - ln(2). The derivative of a constant is zero, so d/dx(ln(2)) = 0. Thus, d/dx (ln(x/2)) = d/dx(ln(x)) = 1/x.

Using Properties of Logarithms

We use the property: ln(a/b) = ln(a) - ln(b). So, ln(x/2) = ln(x) - ln(2). Differentiating both sides, we have: d/dx (ln(x/2)) = d/dx(ln(x)) - d/dx(ln(2)) = 1/x - 0 = 1/x. Thus, the derivative of ln(x/2) is 1/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(x/2).

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(x/2), we generally use f⁽ⁿ⁾(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 approaches 0, the derivative is undefined because ln(x/2) is undefined for x ≤ 0.

When x is 2, the derivative of ln(x/2) = 1/2.

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

• Function: A function involving a logarithm, such as ln(x/2).

• Natural Logarithm: A logarithm to the base e, represented as ln(x).

• Quotient Rule: A method for finding the derivative of a quotient of two functions.

• Chain Rule: A technique for differentiating composite functions.