Derivative of ln Rules

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

The key concepts are mentioned below:

Logarithm Function: ln(x) is the natural logarithm of x.

Chain Rule: A rule used to differentiate composite functions involving ln(x).

Quotient Rule: Used when ln(x) appears in a fraction.

The derivative of ln(x) can be denoted as d/dx (ln x) or (ln x)'. The formula we use to differentiate ln(x) is: d/dx (ln x) = 1/x The formula applies to all x where x > 0.

We can derive the derivative of ln(x) using proofs. To show this, we will use the definition of the natural logarithm and the rules of differentiation. There are several methods we use to prove this, such as:

1. By First Principle
2. Using Chain Rule

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

By First Principle

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

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

As h approaches 0, ln(1 + h/x) / (h/x) approaches 1 (using the limit property of ln), f'(x) = 1/x Hence, proved.

Using Chain Rule

To prove the differentiation of ln(x) using the chain rule, Consider that we have a composite function involving ln(x), such as ln(u(x)), where u(x) is a differentiable function. The chain rule states that: d/dx [ln(u(x))] = (1/u(x)) * u'(x) For the simplest case where u(x) = x, we have: d/dx [ln(x)] = (1/x) * 1 = 1/x

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can provide deeper insights into the behavior of functions. For example, the second derivative can tell us about the concavity of the function.

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), we generally use fⁿ(x) for the nth derivative of a function f(x) which tells us the change in the rate of change.

When x is 0, the derivative is undefined because ln(x) is undefined at x = 0. When x is 1, the derivative of ln(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.

• Natural Logarithm: The natural logarithm is a logarithm to the base e, where e is approximately 2.71828.

• Chain Rule: A fundamental rule in calculus used for differentiating composite functions.

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

• First Principle: The foundational method for deriving the derivative of a function using limits.