Question

Find the derivative of f(x) = 8 divided by x at x = -1.

Answer 1

@ganeshie8 @robtobey if you could assit me i would appreciate your help.

Answer 2

familiar with any formulas ?

Answer 3

would you have to do the difference quotient formula

Answer 4

yes, can you show me the formula which you have ?

Answer 5

f(x+h)-f(x)/h

Answer 6

this is where i dont know how to proceed

Answer 7

what about the limit ?

Answer 8

\[\large f'(x) = \lim \limits_{h\to 0} \dfrac{f(x+h) - f(x)}{h}\]

Answer 9

the limit is -1 in the formula its limit h to 0=f(x+h)-f(x)/h

Answer 10

thats the formula right ?

Answer 11

yes exactly

Answer 12

start by finding the difference `f(x+h) - f(x)`

Answer 13

f(x) = 8/x

f(x+h) = ?

Answer 14

8(x+h) -8/x

Answer 15

yes, put them as single fraction by working the common denominator

Answer 16

x(8(x+h))-8/x

Answer 17

8x(x^2+hx)-8/x

Answer 18

\(\large \dfrac{8}{x+h} - \dfrac{8}{x}\)

Answer 19

multiply the first fraction by x/x
second fraction by (x+h)/(x+h)

Answer 20

\(\large \dfrac{x}{x}\dfrac{8}{x+h} - \dfrac{8}{x}\dfrac{x+h}{x+h}\)

Answer 21

\(\large \dfrac{8x}{x(x+h)} - \dfrac{8(x+h)}{x(x+h)}\)

Answer 22

Now that the denominators are same, you can add/subtract the numerators

Answer 23

\(\large \dfrac{8x - 8(x+h)}{x(x+h)}\)

Answer 24

simplify

Answer 25

8h/x^2+hx

Answer 26

careful about signs *

Answer 27

-8h/x^2+hx

Answer 28

yes, so
f(x+h) - f(x) = -8h/(x^2+hx)

Answer 29

plug this value in your earlier difference quotient formula

Answer 30

\[\large f'(x) = \lim \limits_{h\to 0} \dfrac{f(x+h) - f(x)}{h}\]

Answer 31

\[\large f'(x) = \lim \limits_{h\to 0} \dfrac{-8h/(x^2+hx)}{h}\]

Answer 32

h cancels out ^

Answer 33

\[\large f'(x) = \lim \limits_{h\to 0} -8/(x^2+hx)\]

Answer 34

Since there is no h in the bottom, you can take the limit now

Answer 35

simply plugin h = 0

Answer 36

\[\large f'(x) = \lim \limits_{h\to 0} -8/(x^2+hx)\]

\[\large f'(x) = -8/(x^2+0 \times x)\]

\[\large f'(x) = -8/x^2\]

Answer 37

thats the derivative function ^

Answer 38

plugin x = -1 to get the derivative at x = -1

Answer 39

would it -8 as your derivative

Answer 40

Yep !

Answer 41

thank you for your help its very kind of you to take the the time and effort to assist me. i appreciate your assistance and i hope you have a nice day.

Answer 42

np, you're welcome :) good day !