What is the derivative of log x?

The derivative of log x, where log denotes the natural logarithm (ln), is given by:

f'(x) = 1/x

To understand why this is the case, let’s recall the definition of the natural logarithm. The natural logarithm is the inverse function of the exponential function. That is, if y = ln(x), then x = e^y. To find the derivative of ln(x), we can use implicit differentiation.

Starting with the equation:

x = e^y

we can differentiate both sides with respect to x. The left side becomes 1, and the right side, using the chain rule, becomes e^y (dy/dx):

1 = e^y (dy/dx)

Now, solve for dy/dx:

dy/dx = 1/e^y

Since we know from our original equation that e^y = x, we can substitute back in:

dy/dx = 1/x

This result shows that the derivative of log x (ln x) is indeed 1/x. It’s a crucial result that appears frequently in calculus, especially when dealing with problems involving growth rates or scales.