How to Differentiate ln(u) with Respect to x Using the Chain Rule

d/dx ln(u)

To differentiate ln(u) with respect to x, we use the chain rule

To differentiate ln(u) with respect to x, we use the chain rule. The chain rule states that if we have a composite function, f(g(x)), then its derivative is given by the product of the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function with respect to x.

In this case, ln(u) represents the natural logarithm of u. Therefore, we have a composite function where g(x) = u and f(u) = ln(u). Using the chain rule, we can differentiate ln(u) with respect to x as follows:

d/dx ln(u) = d/dx [f(g(x))]

= f'(g(x)) * g'(x)

First, we find the derivative of f(u) = ln(u) with respect to u, which is given by:

f'(u) = 1/u

Next, we find the derivative of g(x) = u with respect to x. Since u is a function of x, we can use the chain rule again:

g'(x) = du/dx

Finally, substituting the derivatives back into the chain rule equation, we obtain:

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

So, the derivative of ln(u) with respect to x is (1/u) * du/dx.

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