Question

I'm working is session 2 of Differentiation, and I'm clueless how these numbers came out for solving the secant. [(x0)(x0+Δx)/(x0)(x0+Δx)] * [(1\x0+Δx - 1/x0) / Δx] when the previous equation was just the equation after the multiplication symbol?

Answer 1

let's use A and B (easier to type)
you start with \[ \frac{1}{A} + \frac{1}{B}\]
to add the fractions you need a common denominator AB
you can multiply the first fraction by B/B and the second by A/A
\[ \frac{1}{A} \cdot \frac{B}{B} + \frac{1}{B} \cdot \frac{A}{A} = \frac{B}{AB}+\frac{A}{AB}= \frac{B+A}{AB}\]

another way to do this is to multiply the fractions by AB/AB
\[ \frac{AB}{AB} \left( \frac{1}{A} + \frac{1}{B}\right) \]
if we distribute we get
\[ \frac{1}{A} \cdot \frac{AB}{AB}+ \frac{1}{B}\cdot \frac{AB}{AB} \]
which gives
\[ \frac{B}{AB}+ \frac{A}{AB} \]

The idea of multiplying by AB/AB (clearing the denominator) is just a way to add the fractions