Derivative of ln(-5x)

We now explore the derivative of ln(-5x). It is commonly represented as d/dx (ln(-5x)) or (ln(-5x))', and its value is -1/x. The function ln(-5x) has a clearly defined derivative, indicating it is differentiable within its applicable domain. The key concepts are mentioned below: Logarithmic Function: ln(-5x) represents the natural logarithm of -5x. Chain Rule: Used for differentiating composite functions like ln(-5x). Reciprocal Rule: Involved in differentiating ln(x) to obtain 1/x as the derivative.

The derivative of ln(-5x) can be denoted as d/dx (ln(-5x)) or (ln(-5x))'. The formula we use for differentiation is: d/dx (ln(-5x)) = -1/x The formula applies to all x where -5x is positive, that is, when x < 0.

We can derive the derivative of ln(-5x) 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: Using Chain Rule To prove the differentiation of ln(-5x) using the chain rule: Consider f(x) = ln(u) where u = -5x. By the chain rule: d/dx [ln(u)] = (1/u)·(du/dx) Let u = -5x, so du/dx = -5. Substituting these, we get: d/dx (ln(-5x)) = (1/(-5x))·(-5) = -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, consider velocity (first derivative) and acceleration (second derivative) for a moving object. Higher-order derivatives help us understand the behavior of functions like ln(-5x). The first derivative of a function is denoted as f′(x), which indicates how the function changes or its slope at a certain point. The second derivative is derived from the first derivative and is denoted using f′′(x). Similarly, the third derivative, f′′′(x), results from the second derivative, and this pattern continues. For the nth derivative of ln(-5x), we generally use f^(n)(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(-5x) is not defined for non-negative values of x. When x = -1, the derivative of ln(-5x) = -1/(-1) = 1.

Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Logarithmic Function: A function that involves the logarithm, such as ln(-5x). Chain Rule: A rule used to differentiate composite functions. Quotient Rule: A rule used for differentiating functions expressed as a quotient of two functions. Domain: The set of input values (x) for which a function is defined.