Part 1: How to Find the Derivative of the Logarithm Function with Base b – Step-by-Step Guide Part 2: Simplified Derivative of the Logarithm Function with Base b Explained

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 start by assuming y = log base b of x. This can also be written as y = log(x)/log(b).

To find dy/dx, we need to differentiate both sides of the equation with respect to x:

d/dx[y] = d/dx[log(x)/log(b)]

Using the quotient rule, we differentiate the numerator and denominator separately:

d/dx[log(x)] = 1/x (This is the derivative of log(x))

d/dx[log(b)] = 0 (Since log(b) is a constant with respect to x)

Applying the quotient rule:

dy/dx = (d/dx[log(x)] · log(b) – log(x) · d/dx[log(b)]) / (log(b))^2

dy/dx = (1/x · log(b) – log(x) · 0) / (log(b))^2

dy/dx = log(b) / (x · log(b))^2

Simplifying the expression:

dy/dx = log(b) / (x^2 · (log(b))^2)

Therefore, the derivative of log base b of x (denoted as d/dx[log base b of x]) is log(b) / (x^2 · (log(b))^2).

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