Calculating the Derivative of ln(u) with Respect to x using the Chain Rule

d/dx(ln(u))

To compute the derivative of ln(u) with respect to x, denoted as d/dx(ln(u)), we use the chain rule

To compute the derivative of ln(u) with respect to x, denoted as d/dx(ln(u)), we use the chain rule. The chain rule states that if we have a composite function, in this case ln(u), and we want to find its derivative with respect to a different variable, we can do so by taking the derivative of the outer function with respect to the inner variable (u), and multiplying it by the derivative of the inner function with respect to the variable of interest (x).

So, using the chain rule, we have:

d/dx(ln(u)) = (d(ln(u))/du) * (du/dx)

The derivative of ln(u) with respect to u is 1/u. Therefore, we have:

d/dx(ln(u)) = (1/u) * (du/dx)

Now, this expression involves du/dx, which represents the derivative of u with respect to x. If u is a function of x, we can find du/dx by taking its derivative accordingly.

For example, if u = x^2 + 3x, then we can find du/dx as follows:

du/dx = 2x + 3

Substituting this back into our previous expression, we have:

d/dx(ln(u)) = (1/u) * (2x + 3)

And that is the derivative of ln(u) with respect to x. Keep in mind that the specific form of u will affect the calculation, so make sure to adjust accordingly.

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