Derivative of ln(x^3)

We now understand the derivative of ln(x^3). It is commonly represented as d/dx (ln(x^3)) or (ln(x^3))', and its value is 3/x. The function ln(x^3) has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Logarithmic Function: ln(x^3) is a composition of logarithmic and power functions. Chain Rule: Rule for differentiating composite functions like ln(x^3). Natural Logarithm: ln(x) is the logarithm to the base e.

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

We can derive the derivative of ln(x^3) using proofs. To show this, we will use logarithmic identities along with the rules of differentiation. There are several methods we use to prove this, such as: By First Principle Using Chain Rule We will now demonstrate that the differentiation of ln(x^3) results in 3/x using the above-mentioned methods: Using Chain Rule To prove the differentiation of ln(x^3) using the chain rule, Consider f(x) = x^3 and g(x) = ln(x) So, ln(f(x)) = ln(x^3) = 3ln(x) By chain rule: d/dx [ln(f(x))] = f'(x)/f(x) Let’s substitute f(x) = x^3 in the formula, d/dx (ln(x^3)) = d/dx (3ln(x)) = 3 * d/dx (ln(x)) = 3 * (1/x) = 3/x Therefore, the derivative of ln(x^3) is 3/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 ln(x^3). 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(x^3), 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 approaches 0, the derivative is undefined because ln(x) is undefined for x ≤ 0. When x = 1, the derivative of ln(x^3) = 3/1, which is 3.

Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Logarithmic Function: A function involving the logarithm, such as ln(x). Chain Rule: A rule used to differentiate composite functions. Natural Logarithm: The logarithm to the base e, denoted as ln(x). Quotient Rule: A rule for differentiating the quotient of two functions.