Derivative of ln(-x)

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:

Natural Logarithm Function: ln(x) is the logarithm to the base e, where e is the Euler's number.

Chain Rule: A rule for differentiating composite functions.

Negative Sign: The derivative introduces a negative sign due to the nature of the function.

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 chain rule along with the rules of differentiation. There are several methods we use to prove this, such as:

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 Given that f(x) = ln(-x), we write f(x + h) = ln(-(x + h)).

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

Using the limit property ln(1 + y) ≈ y for small y, f'(x) = limₕ→₀ [(h/x) / h] = -1/x Hence, proved.

Using Chain Rule

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

By the chain rule: d/dx [ln(-x)] = (1/u) * (du/dx)

Let’s substitute u = -x, d/dx (ln(-x)) = (1/(-x)) * (-1) = -1/x Hence, proved.

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).

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 f n(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 there.

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 Function: The natural logarithm function is one of the primary logarithmic functions and is written as ln(x).

• Chain Rule: A rule in calculus used to differentiate composite functions.

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

• Undefined: A term used when a function does not have a value at a certain point.