Derivative of ln(3x-2)

We now understand the derivative of ln(3x-2). It is commonly represented as d/dx [ln(3x-2)] or [ln(3x-2)]', and its value is 3/(3x-2). This function 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: A rule for differentiating composite functions. Reciprocal Rule: The derivative of ln(x) is 1/x.

The derivative of ln(3x-2) can be denoted as d/dx [ln(3x-2)] or [ln(3x-2)]'. The formula we use to differentiate ln(3x-2) is: d/dx [ln(3x-2)] = 3/(3x-2) The formula applies to all x where 3x-2>0.

We can derive the derivative of ln(3x-2) using proofs. To show this, we will use the chain rule 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(3x-2) results in 3/(3x-2) using the above-mentioned methods: By First Principle The derivative of ln(3x-2) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of ln(3x-2) using the first principle, we will consider f(x) = ln(3x-2). 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(3x-2), we write f(x + h) = ln(3(x + h) - 2). Substituting these into equation (1), f'(x) = limₕ→₀ [ln(3(x + h) - 2) - ln(3x - 2)] / h Using the properties of logarithms, ln(a) - ln(b) = ln(a/b), f'(x) = limₕ→₀ ln[(3(x + h) - 2)/(3x - 2)] / h = limₕ→₀ ln[1 + (3h/(3x - 2))] / h Using limit properties, limₕ→₀ ln(1 + u)/u = 1 as u→0, f'(x) = 3/(3x-2) Hence, proved. Using Chain Rule To prove the differentiation of ln(3x-2) using the chain rule, We use the formula: Let u = 3x-2, then ln(u) By chain rule: d/dx [ln(u)] = (1/u) * (du/dx) Let’s substitute u = 3x-2, d/dx [ln(3x-2)] = 1/(3x-2) * 3 d/dx [ln(3x-2)] = 3/(3x-2) Using Logarithmic Differentiation We will now prove the derivative of ln(3x-2) using logarithmic differentiation. The step-by-step process is demonstrated below: Consider y = ln(3x-2) Then, dy/dx = d/dx [ln(3x-2)] Using the property of logarithms and chain rule: dy/dx = 1/(3x-2) * d/dx (3x-2) dy/dx = 1/(3x-2) * 3 dy/dx = 3/(3x-2)

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(3x-2). 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(3x-2), 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 = 2/3, the derivative is undefined because ln(3x-2) is undefined at x = 2/3. When x = 1, the derivative of ln(3x-2) = 3/(3*1-2), which equals 3.

Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Chain Rule: A fundamental rule in calculus used to differentiate composite functions. Logarithmic Function: The natural logarithm function, ln(x), which is the inverse of the exponential function. Quotient Rule: A rule used to differentiate functions that are divided by each other. Undefined: A term used when a function does not have a well-defined value at a certain point.