Solving the Derivative of ln(x)

Steps to Solve

We want to find the derivative of ln(x). The derivative of ln(x) is 1/x, and is actually a well-known derivative that most put to memory. However, it’s always useful to know where this formula comes from, so let’s take a look at the steps to actually find this derivative.

To find the derivative of ln(x), the first thing we do is let y = ln(x). Next, we use the definition of a logarithm to write y = ln(x) in logarithmic form. The definition of logarithms states that y = log b (x) is equivalent to b y = x. Therefore, by the definition of logarithms and the fact that ln(x) is a logarithm with base e, we have that y = ln(x) is equivalent to e y = x.

Okay, just a few more steps, and we’ll have our formula! The next thing we want to do is treat y as a function of x, and take the derivative of each side of the equation with respect to x. We use the chain rule on the left hand side of the equation to find the derivative. The chain rule is a rule we use to take the derivative of a composition of functions, and it has two forms.

The left hand side of the equation is e y, where y is a function of x, so if we let f(x) = e x and g(x) = y, then f(g(x)) = e y. Since the derivative of e to a variable (such as e x) is the same as the original, the derivative of f'(g(x)) is e y. Therefore, by the chain rule, the derivative of e y is e y dy/dx. On the right hand side we have the derivative of x, which is 1.

We have (e y) dy/dx = 1. Now, recall that e y = x. We’re going to use this fact to plug x into our equation for e y.

This gives us the equation (x)dy/dx = 1. We’re getting super close now! Are you as excited as I am? We can divide both sides of this equation by x to get dy/dx = 1/x. The last thing is to recall that y = ln(x) and plug this into our equation for y.

Ta-da! Now, we see that d/dx ln(x) = 1/x, and now we know why this formula for the derivative of ln(x) is true. So what’s our solution? The derivative of ln(x) is 1 / x.

Application of Derivative of ln(x)

As we said, the derivative of ln(x) is a well-known derivative that most put to memory. This is because this derivative shows up often in real world applications. Therefore, it’s very useful to know the derivative so that you don’t have to go through the process of finding it every time it comes up.

For example, consider a certain plane that takes off at sea level. The plane’s altitude (in feet) at x minutes can be given by the function h(x) = 2000ln(x).