How to Find the Derivative of the Logarithm Function with Respect to x | Step-by-Step Guide

Derivative of log base b of x

The derivative of the logarithm function with respect to x, where the base is b, can be found using the logarithmic differentiation technique

The derivative of the logarithm function with respect to x, where the base is b, can be found using the logarithmic differentiation technique. Let y = log_b(x), where b > 0 and b ≠ 1.

To find the derivative of y with respect to x, we need to rewrite the logarithm using a change of base formula:

y = log_b(x) = log(x) / log(b)

Taking the natural logarithm (base e) of both sides, we have:

ln(y) = ln(log(x) / log(b))

Using the properties of logarithms, we can simplify this expression:

ln(y) = ln(log(x)) – ln(log(b))

Now, we differentiate both sides of the equation with respect to x:

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

Using the chain rule, the derivative of ln(y) with respect to x becomes:

(1/y) (dy/dx) = (1/log(x)) (1/x)

Rearranging and multiplying both sides of the equation by y, we get:

dy/dx = y / (x log(x))

Substituting y = log_b(x) back into our equation, we find:

dy/dx = log_b(x) / (x log(x))

Hence, the derivative of log base b of x is given by:

dy/dx = log_b(x) / (x log(x))

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