Derivative of ln(1+1/x)

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

The key concepts are mentioned below:

Logarithmic Function: ln(1+1/x).

Chain Rule: Rule for differentiating composite functions like ln(1+1/x).

Reciprocal Rule: Used in simplifying the derivative.

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

The formula applies to all x where x ≠ -1 or 0.

We can derive the derivative of ln(1+1/x) using proofs. To show this, we will use chain and reciprocal rules along with 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(1+1/x) results in -1/(x(x+1)) using the above-mentioned methods:

Using Chain Rule

To prove the differentiation of ln(1+1/x) using the chain rule, Let u = 1+1/x Then ln(1+1/x) = ln(u) By chain rule: d/dx (ln(u)) = (1/u) * (du/dx)

So, we differentiate u with respect to x: du/dx = d/dx (1+1/x) = -1/x²

Therefore, d/dx (ln(1+1/x)) = (1/(1+1/x)) * (-1/x²) = -1/(x(x+1))

Using Quotient Rule

To prove the differentiation of ln(1+1/x) using the quotient rule, Start with the function u = 1+1/x, then differentiate using the quotient rule:

Let u = 1 and v = x, then u/v = 1/x.

d/dx (1/x) = (v * du/dx - u * dv/dx) / v² = (x * 0 - 1 * 1) / x² = -1/x²

Thus, d/dx (ln(1+1/x)) = 1/(1+1/x) * (-1/x²) = -1/(x(x+1))

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+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+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 is -1 or 0, the derivative is undefined because the function ln(1+1/x) has a vertical asymptote there. When x is 1, the derivative of ln(1+1/x) = -1/(1(1+1)) = -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 that involves the logarithm, such as ln(1+1/x).

• Chain Rule: A rule for differentiating composite functions by differentiating the outer and inner functions.

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

• Undefined: A term describing a point where a function does not have a defined value, often leading to division by zero.