Understanding and calculating the derivative of ln(x) using logarithmic properties and the chain rule

Derivative lnx

The derivative of ln(x) can be found using the properties of logarithmic functions and the chain rule of differentiation

The derivative of ln(x) can be found using the properties of logarithmic functions and the chain rule of differentiation.

To begin, let’s write the natural logarithm function ln(x) in exponential form. The natural logarithm of x is defined as the power to which the base e (approximately 2.71828) must be raised to obtain x:

ln(x) = log_e(x) = y (Equation 1)

Rewriting this equation in exponential form:

e^y = x (Equation 2)

We can differentiate both sides of Equation 2 with respect to y:

d/dy (e^y) = d/dy (x)

Using the chain rule, the derivative of e^y with respect to y is simply e^y:

e^y = d/dy (x)

or

e^y = dx/dy

Now, recall that y = ln(x) from Equation 1. Plugging this into the previous equation, we get:

e^ln(x) = dx/dy

Since e^ln(x) is the identity function (e^(ln(x)) = x), the equation simplifies to:

x = dx/dy

In other words, the derivative of ln(x) with respect to x is equal to 1/x:

d/dx (ln(x)) = 1/x

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

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