deriv of ln|x|
To find the derivative of ln|x|, we need to use the properties of the natural logarithm function and the chain rule of differentiation
To find the derivative of ln|x|, we need to use the properties of the natural logarithm function and the chain rule of differentiation.
First, let’s start by rewriting ln|x| as two separate cases:
If x > 0, then ln|x| = ln(x).
If x < 0, then ln|x| = ln(-x).
Now, let's differentiate each case separately:
Case 1: x > 0
To find the derivative of ln(x), we use the chain rule. The chain rule states that if we have a composite function f(g(x)), then the derivative is given by f'(g(x)) * g'(x).
In our case, f(u) = ln(u) and g(x) = x.
Therefore, f'(u) = 1/u (the derivative of ln(u)), and g'(x) = 1 (the derivative of x).
Using the chain rule, we have:
d/dx(ln(x)) = f'(g(x)) * g'(x) = 1/x * 1 = 1/x.
So, if x > 0, the derivative of ln|x| is 1/x.
Case 2: x < 0 To find the derivative of ln(-x), we still use the chain rule. In this case, f(u) = ln(u) and g(x) = -x. Therefore, f'(u) = 1/u (the derivative of ln(u)), and g'(x) = -1 (the derivative of -x). Using the chain rule, we have: d/dx(ln(-x)) = f'(g(x)) * g'(x) = 1/(-x) * (-1) = 1/x. So, if x < 0, the derivative of ln|x| is also 1/x. Therefore, regardless of the sign of x, the derivative of ln|x| is 1/x.
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