The Derivative of ln(x) Explained | A Step-by-Step Guide to Logarithmic Differentiation

Derivative of ln x

To find the derivative of ln(x), we can use the logarithmic differentiation method

To find the derivative of ln(x), we can use the logarithmic differentiation method.

Let y = ln(x). Taking the natural logarithm of both sides, we have:

ln(y) = ln(ln(x))

Now, we can differentiate both sides with respect to x using the chain rule.

On the left side, the derivative of ln(y) with respect to x can be written as (1/y) * (dy/dx).

On the right side, we differentiate ln(ln(x)).

Using the chain rule, the derivative of ln(ln(x)) with respect to x can be written as (1/ln(x)) * (d(ln x)/dx).

The derivative of ln x with respect to x is 1/x.

So, substituting our derivatives on both sides of the equation, we have:

(1/y) * (dy/dx) = (1/ln(x)) * (1/x)

Now, solving for dy/dx, we have:

dy/dx = (1/ln(x)) * (1/x) * y

Substituting y = ln(x), we get:

dy/dx = (1/ln(x)) * (1/x) * ln(x)

Simplifying, we have:

dy/dx = 1/x

Therefore, the derivative of ln(x) is 1/x.

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