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

1/24

Step-by-step explanation:

f(x) = -6/x

I will rewrite this as

f(x) = -6 x^-1

We know the derivative of x^-1 is -1 x^ (-1-1)  or -1 x^-2

df/dx = -6 * -1 x ^ -2

df/dx = 6 x^-2

df/dx = 6/x^2

Evaluated at x=12

df/dx = 6/12^2

         =6/144

        =1/24

1/24

Step-by-step explanation:

We want to find the derivative of f at x=12.

I'm assuming you want to see the formal definition of a derivative approach.

The definition is there:

So we need to find f(x+h) given f(x).

To do this all you have to is replace old input, x, with new input, (x+h).

Let's do that:

.

Let's go to the definition now:

Multiply top and bottom by the least common multiple the denominators of the mini-fractions.  That is, we are going to multiply top and bottom by x(x+h):

Let's cancel the (x+h)'s in the first mini-fraction.

We will also cancel the (x)'s in the second-mini-fraction.

--=+ so I'm rewriting that part:

Distribute (NOT ON BOTTOM!):

Simplify the top (-6x+6x=0):

Simplify the fraction (h/h=1):

Now you can plug in 0 for h because it doesn't give you 0/0:

We want to evaluated the derivative at x=12 so replace x with 12:

Divide top and bottom by 6:

3rd row (abcg) skip a line h

Step-by-step explanation: