Derivative of ln(1-x)

We now understand the derivative of ln(1-x). It is commonly represented as d/dx (ln(1-x)) or (ln(1-x))', and its value is -1/(1-x). The function ln(1-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 natural logarithm of x). Chain Rule: Rule for differentiating ln(1-x) (since it involves a composition of functions).

Reciprocal Function: A function of the form 1/x.

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

We can derive the derivative of ln(1-x) using proofs. To show this, we will use the basic rules of differentiation along with the chain rule

There are several methods we use to prove this, such as:

By First Principle Using Chain Rule We will now demonstrate that the differentiation of ln(1-x) results in -1/(1-x) using the above-mentioned methods:

By First Principle The derivative of ln(1-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(1-x) using the first principle, we will consider f(x) = ln(1-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(1-x), we write f(x + h) = ln(1-(x + h)).

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

Using the property of logarithms ln(a/b) = ln(a) - ln(b), f'(x) = limₕ→₀ ln[(1-x-h)/(1-x)] / h = limₕ→₀ ln[1 - h/(1-x)] / h

Using the approximation ln(1-u) ≈ -u for small u, f'(x) = limₕ→₀ [-h/(1-x)] / h = -1/(1-x)

Hence, proved. Using Chain Rule To prove the differentiation of ln(1-x) using the chain rule, Consider f(x) = 1-x, then ln(1-x) = ln(f(x)).

By the chain rule: d/dx [ln(f(x))] = (1/f(x)) * f'(x)

Let’s substitute f(x) = 1-x, then f'(x) = -1. d/dx (ln(1-x)) = (1/(1-x)) * (-1) = -1/(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(1-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(1-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 approaches 1, the derivative is undefined because ln(1-x) is not defined for x ≥ 1. When x is 0, the derivative of ln(1-x) = -1/(1-0), 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: A logarithm with base e, commonly written as ln(x).

• Chain Rule: A fundamental rule in calculus for finding the derivative of a composite function.

• Quotient Rule: A method used to differentiate functions that are the quotient of two other functions.

• Domain: The set of all possible input values (x-values) for which the function is defined.