d ln U
dU / U
The derivative of ln U with respect to U is 1/U. Thus, applying the chain rule of differentiation, the derivative of ln U with respect to another variable, say x, is given by:
d/dx (ln U) = (d/dU (ln U)) * (dU/dx) = (1/U) * dU/dx
Alternatively, we can rewrite ln U as ln(U(x)) and use the chain rule as follows:
d/dx (ln(U(x))) = (1/U(x)) * dU(x)/dx
Overall, the derivative of ln U with respect to x is equivalent to the derivative of U with respect to x, multiplied by 1/U, or 1/U(x) if U is a function of x.
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