The Derivative of ln(x): Finding the Rate of Change of the Natural Logarithm Function

Derivative of ln

The derivative of ln(x), also known as the natural logarithm function, can be found using the properties of logarithms and the chain rule of differentiation

The derivative of ln(x), also known as the natural logarithm function, can be found using the properties of logarithms and the chain rule of differentiation.

Let’s start with the definition of the natural logarithm: ln(x) = y, where y is the logarithm of x to the base e (the natural logarithm base).

To find the derivative of ln(x), we can differentiate both sides of the equation with respect to x:

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

Using the chain rule, we can rewrite this as:

1/x * dx/dx = dy/dx

Since dx/dx is simply 1, the equation simplifies to:

1/x = dy/dx

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

In differential notation, we can write this as:

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

This means that for any value of x, the slope of the graph of the natural logarithm function at that point is equal to 1 divided by the value of x.

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