Derivative of ln Functions

We now understand the derivative of ln x. It is commonly represented as d/dx (ln x) or (ln x)', and its value is 1/x. The function ln x has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Natural Logarithm Function: ln(x) is the natural logarithm of x. Quotient Rule: Not needed for ln(x), but useful for functions involving divisions. Reciprocal Function: The derivative results in a reciprocal function, 1/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 The formula applies to all x > 0, as ln(x) is defined only for positive x.

We can derive the derivative of ln x using proofs. To show this, we will use the definition of the derivative along with the rules of differentiation. Here are the methods we use to prove this: 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 … (1) Given that f(x) = ln x, we write f(x + h) = ln(x + h). Substituting these into equation (1), f'(x) = limₕ→₀ [ln(x + h) - ln x] / h Using the property of logarithms, ln(a) - ln(b) = ln(a/b), f'(x) = 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 Hence, proved. Using Chain Rule To prove the differentiation of ln x using the chain rule, Consider ln x as u, where x = e^u. Then, d/dx (ln x) = d/du (e^u) · du/dx Since d/du (e^u) = e^u, and du/dx = 1/x, d/dx (ln x) = e^u / x = 1/x Using Logarithmic Differentiation Let y = ln x. Taking the exponential of both sides, e^y = x. Differentiating both sides with respect to x, d/dx (e^y) = d/dx (x) e^y · dy/dx = 1 Substituting back y = ln x, we get, x · dy/dx = 1 dy/dx = 1/x Therefore, the derivative of ln x is 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, 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). 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 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 not defined for x ≤ 0. When x is 1, the derivative of ln x = 1/x, which is 1.

Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Natural Logarithm: The natural logarithm is a logarithm to the base e, where e is approximately equal to 2.71828. Quotient Rule: A rule used for differentiating functions that are the ratio of two differentiable functions. Chain Rule: A rule used for differentiating compositions of functions. Logarithmic Differentiation: A technique used to differentiate functions by taking the logarithm of both sides.