Question

Differentiate \(\ln x,\) for \(x>0,\) and differentiate \(\ln (-x),\) for \(x<0,\) to conclude that \(\frac{d}{d x}(\ln |x|)=\frac{1}{x}\).

Short answer

Answer: The derivative of the function \( \ln |x|\) for any nonzero \(x\) is \(\frac{1}{x}\).

Step-by-step solution

Differentiate \(\ln x\)

To differentiate \(\ln x\) for \(x > 0\), we apply the chain rule: $$\frac{d}{d x} (\ln x) = \frac{1}{x} $$

Differentiate \(\ln(-x)\)

To differentiate \(\ln(-x)\) for \(x < 0\), we apply the chain rule, keeping in mind that the derivative of \(-x\) with respect to \(x\) is \(-1\): $$\frac{d}{d x} (\ln(-x)) = \frac{1}{-x} \cdot (-1) = \frac{1}{x} $$

Combine the results

Now, we combine our results to conclude: For \(x > 0\), \(\frac{d}{d x} (\ln |x|) = \frac{d}{d x} (\ln x) = \frac{1}{x}\). For \(x < 0\), \(\frac{d}{d x} (\ln |x|) = \frac{d}{d x} (\ln(-x)) = \frac{1}{x}\). So, for any nonzero \(x\), the derivative of \( \ln |x|\) is: $$ \frac{d}{d x} (\ln |x|) = \frac{1}{x} $$

Key concepts

Chain Rule

One of the most useful rules in differentiation is the Chain Rule. It allows us to find the derivative of composite functions, where one function is nested within another. The basic idea is to differentiate the outer function and then multiply it by the derivative of the inner function.

• When we differentiate a natural logarithm function like \(\ln x\), we think of it as a simple function with an outer layer (\(\ln \)) and an inner layer (\(x\)).
• For \(\ln(-x)\), the inner function is \(-x\). When applying the chain rule, this involves finding the derivative of the outer function (which is \(\frac{1}{x}\)) and then multiplying it by the derivative of the inner function. In this case, the derivative of \(-x\) is \(-1\).

This careful application of the chain rule ensures accurate differentiation regardless of whether \(x\) is positive or negative by adjusting the expression accordingly.

Logarithmic Differentiation

Logarithmic differentiation is specifically handy for functions involving logarithms and can simplify the process of differentiating complex functions.

• When faced with a problem involving \(\ln |x|\), knowing that the absolute value adds a layer of complexity, we use logarithmic differentiation to tackle the issue smoothly.
• By breaking it down into two separate cases, one for positive \(x > 0\) and another for negative \(x < 0\), we are able to manage the nuances logarithmic functions introduce.

The natural logarithmic function, \(\ln x\), transitions seamlessly into logarithmic differentiation due to its property of producing straightforward derivatives, like \(\frac{d}{dx}(\ln x) = \frac{1}{x}\), which ultimately cover all scenarios when synthesizing \(\ln |x|\). This intermediates complex relationships between \(\ln x\) and \(\ln(-x)\) through logarithmic properties.

Absolute Value

The concept of absolute value plays a critical role, especially when dealing with functions having both positive and negative inputs. \(\ln |x|\) is an excellent example as it requires understanding how absolute value affects differentiation.

• Absolute value, denoted as \(|x|\), essentially strips any negative sign from a number. This impacts how we handle the differentiation since \(x\) can be either positive or negative.
• In the context of \(|x|\), we need to consider two cases: one where \(x > 0\) implying \(\log x\), and the other, \(x < 0\), pointing toward \(\log(-x)\).

The derivative of \(\ln |x|\) unifies these aspects. Whether \(x > 0\) or \(x < 0\), it simplifies to \(\frac{1}{x}\) due to how absolute value negates potential complexities. This is why the derivative is identical across positive and negative realms, demonstrating the uniting power of absolute values.