Find the derivative of the following functions.
y = In |x²-1|

Verified step by step guidance

1. First, recognize that the function y = ln|x² - 1| involves the natural logarithm of an absolute value. The derivative of ln|u| with respect to x is given by (1/u) * (du/dx), where u is a function of x.
2. Identify u as x² - 1. We need to find the derivative of u with respect to x. The derivative of x² - 1 is 2x.
3. Apply the chain rule for differentiation. The derivative of y = ln|x² - 1| is (1/(x² - 1)) * (du/dx), where du/dx is the derivative of x² - 1, which we found to be 2x.
4. Combine the results from the previous steps. The derivative of y = ln|x² - 1| is (2x)/(x² - 1).
5. Ensure the domain of the function is considered. Since we are dealing with ln|x² - 1|, x cannot be ±1 because the expression inside the absolute value becomes zero, which is undefined for the natural logarithm.

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In calculus, the derivative is often denoted as f'(x) or dy/dx, and it provides critical information about the function's behavior, such as its slope and points of tangency.

Natural Logarithm

The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately equal to 2.71828. It is a fundamental concept in calculus, particularly in relation to growth rates and exponential functions. The natural logarithm has unique properties, such as ln(ab) = ln(a) + ln(b), which are useful when differentiating functions involving products or quotients.

Absolute Value Function

The absolute value function, denoted as |x|, returns the non-negative value of x regardless of its sign. In calculus, when differentiating functions that include absolute values, it is essential to consider the piecewise nature of the function, as the derivative may change depending on whether the input is positive or negative. This requires careful analysis of the function's domain and critical points.