The Chain Rule | Differentiating Logarithmic Functions with Respect to x

d/dx(loga(u)) (a is a constant)

To differentiate the logarithm of a function with respect to x, you can use the chain rule

To differentiate the logarithm of a function with respect to x, you can use the chain rule. Let’s calculate d/dx(loga(u)), where a is a constant.

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

In our case, we have y = log_a(u), where u is a function of x. The base of the logarithm is fixed as a constant, so it doesn’t affect the differentiation.

Let’s break down the function into its constituent parts:

f(x) = log_a(x)
g(x) = u(x)

Now, let’s find the derivative of f(x) and g(x):

f'(x) = 1 / (x * ln(a)) — [1] (the derivative of the logarithm base a is 1/(x*ln(a)))

g'(x) = du(x)/dx — [2] (the derivative of u with respect to x)

Now applying the chain rule:

dy/dx = f'(g(x)) * g'(x)

Substituting the expressions for f'(x) and g'(x):

dy/dx = (1 / (g(x) * ln(a))) * du(x)/dx

Since g(x) is u(x):

dy/dx = (1 / (u(x) * ln(a))) * du(x)/dx

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

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