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. We’re going to use this fact to plug x into our equation for e y. The last thing is to recall that y = ln(x) and plug this into our equation for y.We have (e^ y) 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. On the right-hand side we have the derivative of x, which is 1. Therefore, by the chain rule, the derivative of e y is e^y dy/dx. 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. 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. The chain rule is a rule we use to take the derivative of a composition of functions, and it has two forms. We use the chain rule on the left-hand side of the equation to find the derivative. 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. 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. The definition of logarithms states that y = log b (x) is equivalent to b y = x. Next, we use the definition of a logarithm to write y = ln(x) in logarithmic form. To find the derivative of ln(x), the first thing we do is let y = ln(x). 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. The derivative of ln(x) is 1/x and is actually a well-known derivative that most put to memory.
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January 2023
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