What is derivative of ln(x^2)?

The derivative of ln(x²) can be found using a couple of different approaches.

1. Using the Chain Rule:

• The chain rule states that the derivative of a composite function f(g(x)) is f'(g(x)) * g'(x).

• Here, we can consider ln(x²) as f(g(x)) where f(u) = ln(u) and g(x) = x².

• The derivative of f(u) = ln(u) is f'(u) = 1/u. (See: Derivative of Natural Logarithm)

• The derivative of g(x) = x² is g'(x) = 2x. (See: Power Rule)

• Applying the chain rule: d/dx [ln(x²)] = (1/x²) * (2x) = 2/x.

2. Using Logarithmic Properties:

• Before differentiating, we can simplify ln(x²) using the logarithmic property: ln(a^b) = b * ln(a).

• Therefore, ln(x²) = 2 * ln(x).

• Now, differentiating 2 * ln(x) is straightforward: d/dx [2 * ln(x)] = 2 * (d/dx [ln(x)]).

• The derivative of ln(x) is 1/x. (See: Derivative of Natural Logarithm)

• So, d/dx [2 * ln(x)] = 2 * (1/x) = 2/x.

Therefore, the derivative of ln(x²) is 2/x.

Important Note: This result is valid for x ≠ 0. While ln(x²) is defined for both positive and negative x (except 0), ln(x) is only defined for positive x. Therefore when taking the derivative and simplifying, the original domain might change.