Question

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

Answer 1

You can use quotient rule or product rule for this.

Answer 2

if you have to do it by hand, it is going to be some algebra

Answer 3

if not, remember that
\[\frac{d}{dx}[\frac{1}{x}]=-\frac{1}{x^2}\]

Answer 4

you mean this formula?

Answer 5

if you have to compute
\[\lim_{x\to -1}\frac{f(x)-f(-1)}{x+1}\] then there will be algebra involved

Answer 6

how would i enter it in my calculator?

Answer 7

you don't

Answer 8

or you can see how it is the same thing as 8x^(-1) and derive it normally.

Answer 9

my guess is you do cannot use the shortcut rules like the power rule etc, and you have to find it by hand
am i right?

Answer 10

it does not specify how i have to just i just need to find the answer

Answer 11

then there is almost nothing to do
know that the derivative of \(\frac{1}{x}\) is \(-\frac{1}{x^2}\)
multiply by 8 and get \(-\frac{8}{x^2}\) then replace \(x\) by \(-1\)

Answer 12

so the derivative would be 8?

Answer 13

yes

Answer 14

Mike, that formula is the definition of a derivative. From that definition and further analysis, we now have rules for "differentiating" different types of functions and equations. Two of those rules are the Quotient Rule and the Product Rule, either one could be potentially used here.

Here is how the Quotient Rule goes:\[\bf If \ a \ function \ is \ In \ the \ form \ f(x)=\frac{ g(x) }{h(x) } \ then \ f'(x)=\frac{ g'(x)h(x)-g(x)h'(x) }{ [h'(x)]^2 }\] The Product Rule states:\[\bf If \ a \ function \ is \ In \ the \ form \ f(x)=g(x) h(x), then \ f'(x)=g'(x)h(x)+g(x)h'(x) \]

Answer 15

@Mike3h

Answer 16

if you need to show work and are not in a calculus class than you will need to use other methods, but if familiar than power rule will suffice

Answer 17

The derivative is not 8 though, only is it -8 when the condition is x=-1. The derivative itself is -8/x^2