How to Find the Derivative of the Logarithm Function with Base a Using the Chain Rule and Change of Base Formula

Derivative of loga(x)

To find the derivative of the logarithm function with base “a”, we can use the properties of logarithms and the chain rule of differentiation

To find the derivative of the logarithm function with base “a”, we can use the properties of logarithms and the chain rule of differentiation.

Let’s start by considering the function f(x) = log_a(x).

Using the change of base formula, we can express the logarithm with base “a” in terms of natural logarithms:

log_a(x) = ln(x) / ln(a)

Now, let’s find the derivative of f(x) by applying the chain rule:

f'(x) = (1 / x) * (ln(a) / ln(a))

Simplifying this expression, we get:

f'(x) = (ln(a) / (x * ln(a)))

Finally, we can rewrite the derivative as:

f'(x) = ln(a) / (x * ln(a))

Therefore, the derivative of log_a(x) is ln(a) / (x * ln(a)).

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