d/dx [ln|x|] =
To find the derivative of the function f(x) = ln|x|, we can use the chain rule
To find the derivative of the function f(x) = ln|x|, we can use the chain rule.
The chain rule states that if we have a composition of two functions, then the derivative of the composition is the derivative of the outer function multiplied by the derivative of the inner function.
In this case, the outer function is ln(u) and the inner function is |x|. So, we have f(x) = ln(|x|), where the function inside the ln is |x|.
To find the derivative, we need to find the derivative of the outer function and the derivative of the inner function.
The derivative of ln(u) with respect to u is 1/u.
The derivative of |x| with respect to x depends on whether x is positive or negative.
If x > 0, then |x| = x, and the derivative of |x| with respect to x is 1.
If x < 0, then |x| = -x, and the derivative of |x| with respect to x is -1.
Since we are taking the absolute value of x in ln|x|, we need to consider both cases.
So, the derivative of f(x) = ln(|x|) with respect to x is:
d/dx [ln|x|] = d/dx [ln(x)], for x > 0
= 1/x, for x > 0
d/dx [ln|x|] = d/dx [ln(-x)], for x < 0 = 1/(-x) * (-1), for x < 0 = 1/x, for x < 0 Therefore, the derivative of f(x) = ln|x| with respect to x is: d/dx [ln|x|] = 1/x, for x ≠ 0. Note that the derivative is undefined at x = 0 because ln(x) is not defined for x ≤ 0.
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