How to Compute the Derivative of log a(u) using the Chain Rule

d/dx log a(u)

To compute the derivative of log a(u) with respect to x, we can use the chain rule

To compute the derivative of log a(u) with respect to x, we can use the chain rule.

The chain rule states that if we have a composite function y = f(g(x)), then the derivative of y with respect to x is given by dy/dx = f'(g(x)) * g'(x).

So, in our case, let’s consider u as a function of x: u = u(x). Then we can rewrite log a(u) as log a(g(x)), where g(x) = u(x).

Now, let’s compute the derivative step-by-step:

1. First, compute the derivative of the outer function log a(u) with respect to its argument u, using the derivative of the natural logarithm:
d/d(u) log a(u) = 1 / (u * ln(a))

2. Next, compute the derivative of the inner function g(x) = u(x) with respect to x:
d/dx g(x) = d/dx u(x)

3. Finally, we can apply the chain rule:
d/dx log a(u) = d/d(u) log a(u) * d/dx u(x)

Combining the results from step 1 and step 3, we get:
d/dx log a(u) = (1 / (u * ln(a))) * d/dx u(x)

Note that in some cases, the value of u might depend on x implicitly, and in such cases, you might need to use implicit differentiation to compute d/dx u(x).

So, the final answer will depend on the particular form of u(x) and may involve both differential and partial differential equations in some cases.

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