Find the Derivative - d/dx natural log of |x|

Problem

d(ln(x))/d(x)

Solution

1. Identify the function to be differentiated, which is the natural logarithm of the absolute value of x defined for all x≠0

2. Consider the case where x>0 In this case, |x|=x so the function is ln(x) The derivative is 1/x

3. Consider the case where x<0 In this case, |x|=−x Applying the chain rule to ln(−x) gives 1/(−x)⋅(−1) which simplifies to 1/x

4. Combine the results from both cases to find a single expression for the derivative.

Final Answer

d(ln(x))/d(x)=1/x

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