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/x.

The function ln(1/x) has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

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

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

Reciprocal Function: 1/x is the reciprocal of 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/x (or) (ln(1/x))' = -1/x

The formula applies to all x where x > 0.

We can derive the derivative of ln(1/x) using proofs.

To show this, we will use the properties of logarithms along with the rules of differentiation.

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

By First Principle

Using Chain Rule

Using Logarithmic Differentiation

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

Using Chain Rule

To prove the differentiation of ln(1/x) using the chain rule, We use the formula: ln(1/x) = ln(x⁻¹) = -ln(x) Differentiating -ln(x), d/dx [-ln(x)] = -d/dx [ln(x)] d/dx [-ln(x)] = -1/x

Hence, proved.

Using Logarithmic Differentiation

To prove the differentiation of ln(1/x) using logarithmic differentiation,

Consider y = ln(1/x) Taking the derivative, dy/dx = d/dx [ln(x⁻¹)] dy/dx = d/dx [-ln(x)] dy/dx = -1/x

Hence, proved.

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(x + h) + ln(x)] / h

Using the property of logarithms,

ln(a) - ln(b) = ln(a/b), f'(x) = limₕ→₀ ln(x/(x + h)) / h

Converting to exponential form, this becomes, f'(x) = limₕ→₀ ln(1 - h/x) / h

Using the limit property, limₕ→₀ ln(1 - h/x) / h = -1/x f'(x) = -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(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 is 0, the derivative of ln(1/x) is undefined because 1/x is undefined at x = 0.

When x is 1, the derivative of ln(1/x) = -1/x, 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: The logarithmic function is a function related to the inverse of an exponential function, typically represented as ln(x).

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

• Reciprocal Function: A function that is the inverse of another function, typically represented as 1/x.

• Higher-Order Derivative: Refers to the derivative of a derivative, providing insight into the rate of change of the rate of change.