Question

Find the derivative of f(x) = -7/x at x = -3.

Answer 1

I know Liebniz Notation I just can't figure out how to turn the -7/x into x^n

Answer 2

\(\large \frac{d}{dx}\frac{-7}{x}=-7\frac{d}{dx}\frac{1}{x}\)

Answer 3

than you can rewrite it as \(\large x^{-1}\)

Answer 4

then*

Answer 5

that doesn't make any sense, sorry... Can you please try explaining it differently?

Answer 6

you can either use the quotient rule, or factor out the -7.

There's a rule that states that \(\huge \frac{d}{dx}Cf(x)=C*\frac{d}{dx}f(x)\)

Answer 7

basically \(\large \frac{-7}{x}=-7*\frac{1}{x}\)

Answer 8

so the answer is 7/3?

Answer 9

well first we need to find f'(x)

Answer 10

Here's what I did:
f(x)=-7/x at x=-3
f(x)=-7(1/x) at x=-3
f(x)=-7(1/-3)
f(x)=2(1/3)
f(x)=7/3 because the multiple choice answers are all improper fractions

Answer 11

The question asked for the derivative at x=-3

first we find the derivative
next we plug in -3

Answer 12

the derivative would be -7(1/x) at x=-3 or -7(1/-3)

Answer 13

no. the derivative would be \(\large -7\frac{d}{dx}\frac{1}{x}\)

Answer 14

what's d?

Answer 15

d/dx is (leibniz?) notation for derivative

Answer 16

ohh I learned Liebniz Notation as f(x)=x^n becomes nx^(n-1)

Answer 17

that's not what leibniz notation is. that's the power rule.
f'(x) is another way of saying (first) derivative
f(x)||f'(x)
\(x^n\)||\(nx^{n-1}\)

Answer 18

so here we have something of the form \(x^{-1}\)

Answer 19

okay, then what?

Answer 20

apply the power rule...

Answer 21

-1(1)^(-1-1)=-1^-2=-1

Answer 22

why is there a 1 in parentheses? "-1(1)"

Answer 23

to show the step

Answer 24

you're putting in numbers instead of letters and that makes no sense.

Answer 25

x^n -> nx^(n-1)

\(\large x^{-1}=>-1*x^{-1-1}=-x^{-2}\)

Answer 26

using the power rule, we start with x^n. You said we have x^-1. That means x=1 and n=-1. The second half of the power rule is nx^(n-1). Plug in n and x to get -1(1)^(-1-1) or -1^-2 which equals -1

Answer 27

that doesn't mean x=1. it means x=x

Answer 28

You need to brush up on your derivative table

Answer 29

I just learned this...

Answer 30

so if it's -x^-2, then what do I do?

Answer 31

I'm not trying to be harsh, all I'm saying is you'r given a table and a recipe, and you're making this stuff up instead of following the recipe.

Answer 32

you rewrite it in full.

\(\huge -7*\frac{-1}{x^2}\)
\(\huge f'(x) = \frac{d}{dx}=\frac{7}{x^2}\)

Answer 33

so now we plug in x?

Answer 34

into the derivative equation, yeah

Answer 35

so it's 7/9?

Answer 36

yeah

Answer 37

can you maybe try to help me with one more?

Answer 38

post another question. we'll see