What is the swap curve
Finding the equation of tangent line to the curve at the given point
EDIT: Let's clarify a couple of things.
The slope of the secant line between $(a, f(a))$ and $(x,f(x)))$ is $$\frac
The slope of the tangent line at $(a, f(a))$ is $$\lim_
To find the equation of a tangent line, one needs to use the point-slope formula, which I've explained below.
Now, in your case, $f(x) = \sqrt
If we try to evaluate this limit by just plugging in $x = 1$, we get $0/0$, which is a problem (dividing by zero is bad), so we need a new strategy.
Idea: When evaluating the limits of fractions, a good trick is to multiply the top and bottom by the "radical conjugate." So:
Now we can evaluate $$\lim_
The point-slope formula says that a line with slope $m$ that passes through $(x_0, y_0)$ has an equation of the form $$y - y_0 = m(x-x_0).$$
In your case, the tangent line passes through $(1,1)$, so you can plug in $x_0 = 1$, $y_0 = 1$. We'll also have the slope, $m$, from the previous section once we evaluate that limit (which I leave to you to do).