 Let's quickly recap the main ideas from section 2.6 of active calculus on derivatives of inverse functions. Our main goals for this section are to find derivative rules for some important functions that are the inverses or computational opposites of other important functions. Those functions are the natural logarithm function, that's the inverse of the function y equals e to the x, and the arc sine function, which is the inverse of the sine function. We'll state those two rules in a minute, but remember in calculus the process is just as important as the results. So not only the rules themselves, but the method by which we get them is important. So let's focus on that first. To get the derivative of y equals the natural logarithm of x, we use the fact that the natural logarithm is the inverse function for y equals e to the x. That means that if y is equal to ln of x, then e to the y is equal to x. This is actually the definition of the natural logarithm, and this dual relationship between the exponential and logarithmic function is essential for working with either one. Anyway, since e to the y equals x, we can differentiate both sides with respect to x to get y prime times e to the y equals one, and so solving we get y prime equals one over e to the y, and e to the y is equal to x. Therefore, for any positive number x, the derivative of natural logarithm of x is one over x. Now we don't have to re-derive this result every time we want to use it. We just use the result. But study and master this process of getting the rule, because that process can be applied to generate new derivative rules if we need them. For example, we can apply the same process to find the derivative of y equals arc sine of x. When we write y equals arc sine of x, what we mean is that x is the angle whose sine is y, that is sine of y equals x. Differentiating both sides with respect to x here gives us y prime times cosine y equals one, and so y prime is equal to one over cosine of y. Now the fundamental trigonometric identity says that sine squared y plus cosine squared y equals one. So therefore, if sine of y equals x, then sine squared of y equals x squared. And so solving for cosine y, we get cosine y equals the square root of one minus x squared. That allows us to simplify the derivative to y prime equals one over the square root of one minus x squared, and that's the derivative of arc sine of x. Again, the process is at least as important as the result here. To get this derivative formula, we wrote out what we wanted to differentiate, we then used the definition of an inverse function to rewrite that. Then we took a derivative using the chain rule and then simplified the result. This is a general derivation technique that will work with any inverse function. In fact, one general result we have from the section is that if f is differentiable with an inverse function called g, then for any x in the domain of g prime of x, we have that g prime of x is one over f prime of g of x. Again, we don't need to re-derive these derivative rules whenever we need to use them. We just use the results. But the process is important for generating new derivative rules, as you will do in your activities when you develop derivative formulas for the arc tangent function.