d/dx [ln|x|] =
To find the derivative of ln|x|, we need to consider the different cases when x is positive and negative
To find the derivative of ln|x|, we need to consider the different cases when x is positive and negative.
When x is positive (x > 0):
We can rewrite ln|x| as ln(x) because when x is positive, the absolute value of x is equal to x.
Using the chain rule, the derivative of ln(x) is 1/x.
Therefore, when x is positive, d/dx [ln|x|] = 1/x.
When x is negative (x < 0): Since x is negative, the absolute value of x is equal to -x. To differentiate ln|-x|, we will use the chain rule and obtain: d/dx [ln|-x|] = d/dx [ln(-x)] = (1/(-x)) * (-1) = 1/x Therefore, when x is negative, d/dx [ln|x|] = 1/x. Combining the two cases, we conclude that the derivative of ln|x| is d/dx [ln|x|] = 1/x for all x ≠ 0.
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