Which of the following corresponds to the derivative of f(x)=1/(x+2) at x = 2, using the primary definition of a derivative, reduced to its simplest form before taking the limit?

i know the derivative at x=2 is -1/16, but i am confused by the question.

Step-by-step explanation:

The definition of derivative is:

So it asks us not to evaluate the limit part but to simplify the fraction part.

So let's focus on just:

We are given .

So (I just replaced the 's with 's.)

Now since we want to find it a . I'm going to replace my x's with 2:

So instead we will look at:

Let's simplify some of the addition that we can in the denominators of the mini-fractions:

Now division by can be written as multiplication by :

Let's combine the fractions inside the ( ).

I will multiply the first fraction by .

I will multiply the second fraction by .

We are going to do this so we have the same denominator:

Now we have the same denominator inside the ( ) and can combine those fractions:

Let's simplify the numerator in the ( ).

Now you should see a common factor to cancel. That is we have that . So we can write that:

So the answer we are looking for is:

E

Step-by-step explanation:

[f(2+h) - f(2)]/h

f(2+h) = 1/(2+h+2) = 1/(4+h)

f(2) = 1/(2+2) = 1/4

[1/(4+h) - 1/4] ÷ h

[4 - (4+h)] ÷ [4h(4+h)]

-h ÷ [4h(4+h)]

-1/[4(4+h)]

d is the correct answer

step-by-step explanation:

x= 8 because 3x = 3 * 8, and 3*8 = 24