Derivative of 6lnx

We now understand the derivative of 6ln x. It is commonly represented as d/dx (6ln x) or (6ln x)', and its value is 6/x. The function 6ln 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).

Constant Multiple Rule: When differentiating a constant multiplied by a function.

Derivative of ln(x): The derivative of ln(x) is 1/x.

The derivative of 6ln x can be denoted as d/dx (6ln x) or (6ln x)'. The formula we use to differentiate 6ln x is: d/dx (6ln x) = 6/x The formula applies to all x where x > 0.

We can derive the derivative of 6ln x using proofs. To show this, we will use the rules of differentiation.

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

1. By First Principle
2. Using Constant Multiple Rule

We will now demonstrate that the differentiation of 6ln x results in 6/x using the above-mentioned methods:

By First Principle

The derivative of 6ln x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of 6ln x using the first principle, we will consider f(x) = 6ln 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) = 6ln x, we write f(x + h) = 6ln (x + h).

Substituting these into equation (1), f'(x) = limₕ→₀ [6ln(x + h) - 6ln x] / h = limₕ→₀ [6(ln(x + h) - ln x)] / h = 6 · limₕ→₀ [ln(x + h) - ln x] / h

Using the property of logarithms: ln a - ln b = ln(a/b), = 6 · limₕ→₀ ln((x + h)/x) / h = 6 · limₕ→₀ ln(1 + h/x) / h

As h approaches 0, ln(1 + h/x) ≈ h/x, f'(x) = 6 · (1/x) = 6/x

Hence, proved.

Using Constant Multiple Rule

To prove the differentiation of 6ln x using the constant multiple rule, We use the formula: 6ln x = 6 · ln x The derivative of ln x is 1/x. So, the derivative of 6ln x is: d/dx (6ln x) = 6 · d/dx (ln x) = 6 · (1/x) = 6/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 6ln(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 6ln(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 is undefined because ln(x) is undefined there. When x is 1, the derivative of 6ln x = 6/1, which is 6.

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

• Logarithmic Function: A logarithmic function is a function of the form ln(x), where ln is the natural logarithm.

• Constant Multiple Rule: A differentiation rule stating that the derivative of a constant times a function is the constant times the derivative of the function.

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

• Quotient Rule: A method for finding the derivative of a ratio of two functions.