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The derivative of the natural logarithm function is 1/x.

The natural logarithm function, denoted as ln(x), is the inverse of the exponential function e^x. Its derivative can be found using the chain rule and the fact that the derivative of e^x is e^x.

Let y = ln(x). Taking the derivative of both sides with respect to x, we get:

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

Using the chain rule, we can rewrite this as:

dy/dx = d/dy(e^y) * d/dx(ln(x))

Since e^y = x, we can simplify this to:

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

Now we need to find the derivative of ln(x). We can use the fact that ln(x) is the inverse of e^x, so ln(x) = y if and only if x = e^y. Taking the derivative of both sides with respect to x, we get:

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

1 = e^y * dy/dx

dy/dx = 1/e^y

Substituting e^y = x, we get:

dy/dx = 1/x

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

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