Derivative of ln|x|

The derivative of ln|x| is commonly represented as d/dx (ln|x|) or (ln|x|)', and its value is 1/x. The natural logarithm function ln|x| has a well-defined derivative, indicating it is differentiable within its domain, except at x = 0. The key concepts are mentioned below: Logarithm Function: ln|x| is the natural logarithm of the absolute value of x. Absolute Value: The absolute value function ensures that ln|x| is defined for both positive and negative x. Reciprocal Function: The derivative 1/x represents the reciprocal of x.

The derivative of ln|x| can be denoted as d/dx (ln|x|) or (ln|x|)'. The formula we use to differentiate ln|x| is: d/dx (ln|x|) = 1/x This formula applies to all x ≠ 0.

We can derive the derivative of ln|x| using proofs. To show this, we will use the properties of logarithms and the rules of differentiation. There are several methods we use to prove this, such as: By First Principle Using Chain Rule By First Principle The derivative of ln|x| can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of ln|x| using the first principle, consider f(x) = ln|x|. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h Given that f(x) = ln|x|, we write f(x + h) = ln|x + h|. Substituting these into the equation, f'(x) = limₕ→₀ [ln|x + h| - ln|x|] / h = limₕ→₀ ln(|x + h|/|x|) / h = limₕ→₀ ln(1 + h/x) / h Using the approximation ln(1 + u) ≈ u for small u, f'(x) = limₕ→₀ (h/x) / h = 1/x Thus, f'(x) = 1/x, hence proved. Using Chain Rule To prove the differentiation of ln|x| using the chain rule, Consider the function ln|x| = ln(x) for x > 0 and ln|x| = ln(-x) for x < 0. For x > 0, d/dx (ln(x)) = 1/x. For x < 0, d/dx (ln(-x)) = -1/x x d/dx(-x) = 1/x. Thus, for both positive and negative x, d/dx (ln|x|) = 1/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, consider a scenario 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|. 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|, 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|x| is not defined at x = 0. When x is 1, the derivative of ln|1| = 1/1, which is 1.

Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Logarithm Function: ln|x| is the natural logarithm of the absolute value of x. Absolute Value: The absolute value function ensures that ln|x| is defined for both positive and negative x. Chain Rule: A rule for differentiating compositions of functions, used when differentiating functions like ln|x|. Quotient Rule: A rule for differentiating ratios of functions, applied in finding derivatives of expressions like ln|x|/x.