Question

Find the derivative of f(x) = negative 6 divided by x at x = 12 and Find the derivative of f(x) = 6x + 2 at x = 1.

Answer 1

\[f(x)=\frac{-6}{x}\]
are you suppose to find f' using the definition of derivative?

Answer 2

Do I plug it in

Answer 3

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

Answer 4

Im just confused since my textbook teaches it like that but doesnt really explain

Answer 5

\[f(x)=\frac{-6}{x} \\ f(x+h)=\frac{-6}{x+h}\]

Answer 6

plug those into the definition above

Answer 7

\[f'(x)=\lim_{h \rightarrow 0}\frac{\frac{-6}{x+h}-\frac{-6}{x}}{h} =\lim_{h \rightarrow 0}\frac{1}{h}(\frac{-6}{x+h}-\frac{-6}{x}) \]
combine those fractions inside the ( )

Answer 8

we are trying to get a factor of h in the numerator that will cancel with that factor of h in the denominator

Answer 9

how do i know to put it under x+h

Answer 10

what?

Answer 11

Nevermind

Answer 12

I'm using the definition of derivative
\[f'(x)=\lim_{h \rightarrow 0}\frac{f(x+h)-f(x)}{h}\]
all I did was replace f(x+h) and f(x)

Answer 13

f(x)=-6/x
so
f(x+h)=-6/(x+h)

Answer 14

so I just put those two things in

Answer 15

Oh ok I see

Answer 16

so you need to combine the fractions in the ( )

Answer 17

\[f'(x)=\lim_{h \rightarrow 0}\frac{\frac{-6}{x+h}-\frac{-6}{x}}{h} =\lim_{h \rightarrow 0}\frac{1}{h}(\frac{-6}{x+h}-\frac{-6}{x}) \]

Answer 18

we are trying to get a factor of h on top so we can cancel the h on bottom

Answer 19

So the h and x cancel

Answer 20

not the x oops

Answer 21

so 6/12?

Answer 22

@freckles

Answer 23

no right

Answer 24

have you combined the fractions in the ( ) yet

Answer 25

(-6/x)- (-6/x)?

Answer 26

what happen to the h?

Answer 27

i thought i ws supposed to cancel it

Answer 28

you have -6/(x+h)-(-6/x)

Answer 29

with what?

Answer 30

h don't just go away

Answer 31

the whole reason we are combine the fractions is so we can cancel the factor h on the outside of the ( )

Answer 32

so what is the valye of h

Answer 33

is the answer 2

Answer 34

why won;t you try to combine the fractions?

Answer 35

h is a variable

Answer 36

oh okay let me try

Answer 37

umm -6/h plus 6

Answer 38

\[f'(x)=\lim_{h \rightarrow 0}\frac{\frac{-6}{x+h}-\frac{-6}{x}}{h} =\lim_{h \rightarrow 0}\frac{1}{h}(\frac{-6}{x+h}-\frac{-6}{x}) \\ =\lim_{h \rightarrow 0} \frac{1}{h}(\frac{-6(x)-(-6)(x+h)}{(x)(x+h)}) \\ \]