(d/dx) ln |x| =
To find the derivative of ln|x| with respect to x, we will use the properties of logarithmic differentiation
To find the derivative of ln|x| with respect to x, we will use the properties of logarithmic differentiation.
First, let’s rewrite ln|x| using absolute value as:
ln|x| = ln(x), when x > 0,
ln|x| = ln(-x), when x < 0.
Now, we will find the derivative in two parts, one for when x > 0 and one for when x < 0.
When x > 0:
We can differentiate ln(x) using the property that d/dx ln(u) = (1/u) * du/dx. In this case, u = x, so:
(d/dx) ln(x) = (1/x) * d/dx(x) = (1/x) * 1 = 1/x.
When x < 0: We differentiate ln(-x) using the chain rule. Let u = -x, so du/dx = -1. Therefore: (d/dx) ln(-x) = (1/u) * du/dx = (1/(-x)) * (-1) = 1/x. So, regardless of the sign of x, the derivative of ln|x| is 1/x.
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