Welcome

This is the first post here, mostly to confirm LaTeX math and a collapsible word-problem layout both render correctly.

A derivative

The derivative of \(f(x) = x^2\) from first principles:

\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h} = 2x \]

A word problem

Problem

A train leaves the station traveling at 60 mph. Two hours later, a second train leaves the same station on the same track traveling at 90 mph. How long after the second train departs does it catch up to the first?

Solution

Let \(t\) be the number of hours after the second train departs.

• First train's position at that time: \(60(t + 2)\)
• Second train's position at that time: \(90t\)

The second train catches up when the positions are equal:

\[ 90t = 60(t + 2) \implies 30t = 120 \implies t = 4 \]

The second train catches up 4 hours after it departs.