Understanding the Derivative of ln(U) with Respect to a Variable | Step-by-Step Explanation and Application of the Chain Rule

d ln U

The expression “d ln U” involves the derivative of the natural logarithm of a function U with respect to some variable

The expression “d ln U” involves the derivative of the natural logarithm of a function U with respect to some variable. Let’s break it down step by step.

1. Natural logarithm (ln):
The natural logarithm, denoted as ln(x), is the inverse function of the exponential function. It is defined for positive real numbers and gives the power to which the base number ‘e’ (approximately 2.718) must be raised to obtain the argument. In other words, it calculates the exponent that results in a given number. For example, ln(e) = 1 and ln(10) ≈ 2.3026.

2. Derivative (d):
The derivative of a function represents the rate at which the function changes with respect to its variable. It measures the instantaneous rate of change of the function at a given point. The notation “d” is often used to indicate differentiation.

3. d ln U:
In this expression, U represents a function, and we are looking to find the derivative of ln(U) with respect to some variable. This implies that U itself is a function of that variable.

To calculate the derivative of ln(U), we use the chain rule. The chain rule states that the derivative of the composition of two functions f(g(x)) is given by the derivative of the outer function f evaluated at the inner function multiplied by the derivative of the inner function g.

So, using the chain rule, the derivative of ln(U) with respect to the variable (let’s call it x) is:

d ln(U)/dx = (1/U) * dU/dx

Where dU/dx represents the derivative of the function U with respect to x.

This expression tells us that the derivative of ln(U) is equal to the derivative of U divided by U itself.

Remember to properly evaluate the derivative of U with respect to x based on the specific form of the function U and the given variable.

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