Question

Help with a quick derivative problem?

Find the derivative of f(x) = \(\dfrac 5x\) at x = -1.

Answer 1

\[f(x)=\frac{ 5 }{x}=5x ^{-1}\]

Answer 2

Now you can use the power rule

Answer 3

\(\large\color{black}{ \displaystyle f(x)=5(x)^{-1} }\)

\(\large\color{black}{ \displaystyle f'(x)=\color{red}{(-1)\times }5(x)^{-1\color{red}{-1}} }\)

Answer 4

simplify that...

Answer 5

I'm not sure if I'm familiar with the power rule

Answer 6

oh, I can introduce you then:) ((( I am using a d/dx notation to denote a derivative, or a notation of f'(x) for a derivative of some function f(x) )))

\(\large\color{black}{ \displaystyle \frac{\rm d }{ {\rm d}x} \left(x^{\rm n}\right)={\rm n}\cdot x^{{\rm n}-1} }\) (this is the power rule)

Answer 7

the above rule applies to any power n, besides n=0.

Answer 8

Okay, that doesn't seem too bad :)

Answer 9

yes, not bad at all.

can you write the power rule for x^3 ?

Answer 10

\(\large\color{black}{ \displaystyle \frac{\rm d }{ {\rm d}x} \left(x^{\rm 3}\right)=~? }\)

what would that be?

Answer 11

\(3*x^2\)?

Answer 12

yes, \(3x^2\)

Answer 13

Okay, not too bad at all :)
Seem pretty easy right now actually :3

Answer 14

\(\large\color{black}{ \displaystyle \frac{\rm d }{ {\rm d}x} \left(x^{\rm 3}\right)={\rm 3} x^{2} }\)

(that line says: "the derivative of x^3 is equal to 3x^2" )

Answer 15

yes

Answer 16

now, go ahead and do the power rule for \(x^{-1}\):

\(\large\color{black}{ \displaystyle \frac{\rm d }{ {\rm d}x} \left(x^{\rm -1}\right) }\)

Answer 17

Just to make sure, are you expected to take the derivative by the limit definition? That would explain why you don't know the power rule.

Answer 18

oh... I haven't thought that way because i read "a `quick` derivative problem" :)

Answer 19

It doesn't tell us a specific way to do it, just wants us to find the answer somehow

Answer 20

ok, so then using power rule would be simpler

Answer 21

\(\large\color{black}{ \displaystyle \frac{\rm d }{ {\rm d}x} \left(x^{\rm \color{red}{-1}}\right)={\rm \color{red}{(-1)}}\cdot x^{{\rm \color{red}{-1}}-1} }\)

Answer 22

now, for any coefficient c in front of \(x^{\rm n}\), (of course c\(\ne\)0) the derivative is:

\(\large\color{black}{ \displaystyle \frac{\rm d }{ {\rm d}x} \left({\rm c}\cdot x^{\rm n}\right)={\rm c}\cdot {\rm n}\cdot x^{{\rm n}-1} }\)

Answer 23

in your case the power of x - the n, is -1. & the coefficient c in front is 5.

Answer 24

\(-5x^{-2}\)

Answer 25

yup

Answer 26

or, \(\large\color{black}{ \displaystyle \frac{ -5 }{ x^2} }\)

Answer 27

now, you need to plug in -1 for x.

Answer 28

so, 5?

Answer 29

no exactly:

Answer 30

\(\large\color{black}{ \displaystyle \frac{-5}{x^2} }\) is the derivative.

plug in -1 for x

\(\large\color{black}{ \displaystyle \frac{-5}{(-1)^2} }\)

and you get ?

Answer 31

-5/1, so 5!

Answer 32

*-5

Answer 33

yes, -5

(but don't put the exclamation mark, because then it is a factorial of 5 times -1) jk

Answer 34

f`(-1)=-5

Answer 35

Saying , that the slope of the function f(x)=5/x, at x=-1 is equivalent to -5.