Concise Derivative of Logarithm Function with Base b: Chain Rule and Change of Base Formula

Derivative of log base b of x

To find the derivative of the logarithm function with base b, we can use the chain rule

To find the derivative of the logarithm function with base b, we can use the chain rule.

Let’s define the function f(x) = log base b of x, where b is a positive constant.

According to the chain rule, the derivative of f(x) with respect to x is given by:

f'(x) = (d/dx) [log base b of x]

Now, we can rewrite the logarithm function using the change of base formula. The change of base formula states that log base b of x is equal to the natural logarithm of x divided by the natural logarithm of b:

f(x) = log base b of x = ln(x) / ln(b)

Using this expression, we can find the derivative of f(x) as follows:

f'(x) = (d/dx) [ln(x) / ln(b)]

To simplify, we can use the quotient rule, which states that the derivative of (u/v) is given by:

(du/dx * v – u * dv/dx) / v^2

Applying the quotient rule to f(x), let u = ln(x) and v = ln(b):

f'(x) = (d/dx) [ln(x) / ln(b)]
= [(d/dx)(ln(x)) * ln(b) – ln(x) * (d/dx)(ln(b)))] / ln(b)^2
= [((1/x) * ln(b)) * ln(b) – ln(x) * (0)] / ln(b)^2
= ln(b) / x * ln(b) / ln(b)^2
= 1 / x * ln(b)

Therefore, the derivative of log base b of x is equal to 1 divided by the natural logarithm of b multiplied by x:

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

This derivative represents the rate of change of the logarithm function with respect to x.

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