Derivative of logn

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

The key concepts are mentioned below:

Logarithmic Function: (logn = ln(n)).

Power Rule: Rule for differentiating logn (since it consists of ln(n)).

Reciprocal Function: 1/n.

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

We can derive the derivative of logn using proofs. To show this, we will use the logarithmic identities along with the rules of differentiation.

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

1. By First Principle
2. Using Chain Rule
3. Using Power Rule

We will now demonstrate that the differentiation of logn results in 1/n using the above-mentioned methods:

By First Principle

The derivative of logn can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of logn using the first principle, we will consider f(n) = logn.

Its derivative can be expressed as the following limit. f'(n) = limₕ→₀ [f(n + h) - f(n)] / h … (1)

Given that f(n) = logn, we write f(n + h) = log(n + h).

Substituting these into equation (1), f'(n) = limₕ→₀ [log(n + h) - logn] / h = limₕ→₀ [ln((n + h)/n)] / h = limₕ→₀ [ln(1 + h/n)] / h Using the formula ln(1 + u) ≈ u for small u, f'(n) = limₕ→₀ (h/n) / h = 1/n

Hence, proved.

Using Chain Rule

To prove the differentiation of logn using the chain rule, We use the formula: logn = ln(n) Consider g(n) = ln(n)

The derivative is 1/n Thus: d/dn (logn) = 1/n

Using Power Rule

We will now prove the derivative of logn using the power rule. The step-by-step process is demonstrated below:

Here, we use the formula, logn = ln(n)

Given that, u = n

Using the power rule formula: d/dn (u^1) = 1 * u^(1-1) 1/n Thus: d/dn (logn) = 1/n

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 logn.

For the first derivative of a function, we write f′(n), 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′′(n) Similarly, the third derivative, f′′′(n) is the result of the second derivative and this pattern continues.

For the nth Derivative of logn, we generally use fⁿ(n) for the nth derivative of a function f(n) which tells us the change in the rate of change. (continuing for higher-order derivatives).

When n is 0, the derivative is undefined because logn is undefined at that point. When n is 1, the derivative of logn = 1/1, which is 1.

• Derivative: The derivative of a function indicates how the given function changes in response to a slight change in n.

• Logarithmic Function: The logarithmic function is written as logn or ln(n).

• Reciprocal Function: A function that is the reciprocal of n, represented as 1/n.

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

• Undefined: A condition where a function does not exist at certain points, such as logn at n = 0.