Question

In Session 1, when finding the slope of the tangent to f(x)=1/x he algebraically gets that ((1/x0+x)-(1/x0))/x) equals (1/x)((x0-(x0+x)/(x0 + x)x0). I’ve watched and rewatched his explanation and read the explanation in the notes .pdf about ten times, but I’m not seeing how he got there. Could someone explain this to me, perhaps differently than the video or the notes?

Answer 1

Could you please share the minute:seconds mark this is presented?

Answer 2

He's just putting the numerator, which is the difference of two fractions, over
a common denominator, namely the product of the two denominators.
So\[1 \div (x + h) - 1 \div x\] (I'm writing x for x-zero and h for delta x) can be put over the common denominator x * (x + h) if you multiply the left half by h and the right half by (x + h). This gives you \[x \div (x * (x + h)) - (x + h) \div (x * (x + h))\]
which you can now put over the common denominator and get \[(x - (x + h)) \div (x * (x + h)) = -h \div (x * (x + h))\]
But you already have an initial 1 / h, which cancels with the h in the numerator above, and so as h goes to zero, the above has a limit of \[-1 \div x^2\]

Answer 3

So he just figured out a common denominator and multiplied both sides by 1 essentially to simplify, right? I think missing the intermediate step was what was throwing me. Thanks.