Find the Derivative of Logarithm with Base a Using the Chain Rule

Derivative of log(. a)x

To find the derivative of log base a (x), you can use the chain rule

To find the derivative of log base a (x), you can use the chain rule. The chain rule states that if you have a function of the form f(g(x)), where f(u) is a function and g(x) is another function, then the derivative of f(g(x)) is f'(g(x)) * g'(x).

In this case, your function is log base a (x), which can also be written as log(x)/log(a).

Let’s denote f(u) = log(u) and g(x) = x/log(a). Then we have:

f(g(x)) = f(x/log(a)) = log(x/log(a))

To find the derivative, we need to find the derivative of f(g(x)) with respect to x.

Using the chain rule, we can write:

(f(g(x)))’ = f'(g(x)) * g'(x)

The derivative of f(u) = log(u) with respect to u is 1/u (this is a basic rule of logarithmic differentiation).

So, f'(u) = 1/u.

The derivative of g(x) = x/log(a) with respect to x is 1/log(a) (the derivative of x is 1, and the derivative of log(a) is 0).

Substituting these derivatives into the chain rule equation, we have:

(f(g(x)))’ = (1/(x/log(a))) * (1/log(a))

Simplifying the expression, we get:

(f(g(x)))’ = 1/(x * log(a) * log(a))

Therefore, the derivative of log base a (x) is 1/(x * log(a) * log(a)).

Alternatively, you can also use rules of logarithms to derive the same result. The derivative of log base a (x) can be written as:

(d/dx) log(x)/log(a) = (1/(x * log(a))) * (d/dx) log(x)

The derivative of log(x) with respect to x is 1/x.

Therefore, the derivative of log base a (x) is 1/(x * log(a) * log(a)).

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