Question

Can someone walk me through the arithmetic of this derivative problem? (Attached)

Answer 1

beat the clock. good thing you will be able to do this in your head in week 3

Answer 2

need time, so much time...

Answer 3

wow!

Answer 4

but i messed up of course

Answer 5

polpak i was in race and polpak is gonna win because he does everything with accuracy

Answer 6

\[f(x) = -\frac{2}{x^2}\]\[\implies \lim_{h\rightarrow 0} \frac{f(x+h) - f(x)}{h}\]\[= \lim_{h\rightarrow 0}\frac{-\frac{2}{(x+h)^2} + \frac{2}{x^2}}{h}\]\[= \lim_{h\rightarrow 0}\frac{-\frac{2}{(x+h)^2} + \frac{2}{x^2}}{h}\cdot \frac{x^2(x+h)^2}{x^2(x+h)^2}\]\[= \lim_{h\rightarrow 0}\frac{-2x^2 + 2(x+h)^2}{hx^2(x+h)^2}\]\[= \lim_{h\rightarrow 0}\frac{-2x^2 + 2(x^2 + 2xh + h^2)}{hx^2(x+h)^2}\]\[= \lim_{h\rightarrow 0}\frac{-2x^2 + 2x^2 + 4xh + 2h^2}{hx^2(x+h)^2}\]\[= \lim_{h\rightarrow 0}\frac{4xh + 2h^2}{hx^2(x+h)^2}\]\[= \lim_{h\rightarrow 0}\frac{\cancel{h}(4x + 2h)}{\cancel{h}x^2(x+h)^2}\]\[=\frac{4x}{x^4} = \frac{4}{x^3}\]

Answer 7

our lady of the latex. joke is that next week you will day
\[-\frac{2}{x^2}=-2x^{-2}\] so
\[f'(x)=-2\times -2x^{-2-1}=4x^{-3}=\frac{4}{x^3}\] and yes you messed up!

Answer 8

Not everything!

Answer 9

But lots of things, yes.

Answer 10

i forgot to distribute the two :(

Answer 11

lol slow and steady wins the race.

Answer 12

i only multiplied first term by 2
and the rest ignored :(

Answer 13

Yeah I panicked for a second when I saw a 2 up top, then realized your mistake.

Answer 14

the turtle wins this time but the rabbit will win someday

Answer 15

all easier if you just show that
\[(\frac{1}{f})'=\frac{-f'}{f^2}\] and then you are done quickly

Answer 16

now i feel like writing it without clearing the fractions. hmmm

Answer 17

So what exactly is the difference quotient for the first part of the question?

Answer 18

\[= \lim_{h\rightarrow 0}\frac{\cancel{h}(4x + 2h)}{\cancel{h}x^2(x+h)^2} \]

the part right before he plugged in 0 for h

Answer 19

just take lim and you have it

Answer 20

i mean take the symbol lim away

Answer 21

IYou can see that the third option is the same as that if you factor out the 2 from the numerator.