Derivative of ln(2/x)

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

The key concepts are mentioned below:

Logarithmic Function: ln(x) represents the natural logarithm of x.

Chain Rule: Rule for differentiating ln(2/x) (since it involves a composition of functions).

Reciprocal Rule: Involves differentiating 1/x.

The derivative of ln(2/x) can be denoted as d/dx [ln(2/x)] or [ln(2/x)]'.

The formula we use to differentiate ln(2/x) is: d/dx [ln(2/x)] = -1/x (or) [ln(2/x)]' = -1/x

The formula applies to all x where x > 0.

We can derive the derivative of ln(2/x) 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:

1. By First Principle
2. Using Chain Rule
3. Using Quotient Rule

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

By First Principle

The derivative of ln(2/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(2/x) using the first principle, we will consider f(x) = ln(2/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(2/x), we write f(x + h) = ln(2/(x + h)).

Substituting these into equation (1), f'(x) = limₕ→₀ [ln(2/(x + h)) - ln(2/x)] / h = limₕ→₀ [ln((2/x) / (2/(x + h)))] / h = limₕ→₀ [ln((x + h)/x)] / h

We now use the logarithmic property ln(a/b) = ln(a) - ln(b). f'(x) = limₕ→₀ [ln(1 + h/x)] / h

Using the limit property ln(1 + u) ≈ u when u is small, f'(x) = limₕ→₀ (h/x) / h = limₕ→₀ 1/x f'(x) = -1/x

Hence, proved.

Using Chain Rule

To prove the differentiation of ln(2/x) using the chain rule, We use the formula: ln(2/x) = ln(2) - ln(x)

The derivative of ln(x) is 1/x, and the constant ln(2) differentiates to 0. d/dx [ln(2/x)] = d/dx [ln(2) - ln(x)] d/dx [ln(2)] = 0 and d/dx [-ln(x)] = -1/x Therefore, d/dx [ln(2/x)] = -1/x

Using Quotient Rule

We will now prove the derivative of ln(2/x) using the quotient rule. The step-by-step process is demonstrated below: Here, we use the formula, ln(2/x) = ln(2) - ln(x)

The derivative of a constant is 0, and the derivative of ln(x) is 1/x. Using these rules: d/dx [ln(2/x)] = d/dx [ln(2) - ln(x)]

Since d/dx [ln(2)] = 0, d/dx [-ln(x)] = -1/x Thus: d/dx [ln(2/x)] = -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(2/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 n^th Derivative of ln(2/x), we generally use fⁿ(x) for the n^th 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 is undefined because ln(2/x) is undefined there. When x = 1, the derivative of ln(2/x) = -1/1, which is -1.

• 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 the natural logarithm, often represented as ln(x).

• Chain Rule: A rule used to differentiate compositions of functions.

• First Derivative: It is the initial result of a function, which gives us the rate of change of a specific function.

• Undefined: A condition where a function does not have a value at a certain point, such as ln(2/x) at x = 0. ```