Calculus help! Which is equal to what?

Question

When given f(x) = 2/x, which of the following is equivalent to (f(x+h)-f(x)) / h ?

A) -2 / x^(2)

B) -x^(2) / x^(2) + xh

C) -2 / x^(2) + xh

D) 2 / x^(2)

E) -h / x^(2) + xh

Answer 1

The correct answer is C and I will show you why!

Let the difference quotient/derivative by defined by (f(x+h)-f(x)) / h. We have f(x) given already. Now lets find f(x+h).

f(x+h) = 2/(x+h). Lets plug everything we have in !

(f(x+h)-f(x)) / h | 2/(x+h) - (2/x) / h | [2x-2(x+h)/x(x+h)] / h | [2x-2x-2h / xh(x+h)] = -2 / (x²+xh)

I hope this helps!

Answer 2

If C means -2/(x^2+xh) then it's C.

Because it is multiple choice, you can use the answer choices to help you narrow down the math. Because I knew that the derivative of f(x) was -2/x^2, I knew that the limit of my answer choice as h goes to 0 had to be that. This is because we are calculating a difference quotient. The limit of (f(x+h)-f(x)) / h as h goes to 0 will be the derivative of f.

Any answer choice that does not have this same limit is wrong. Thus, E is wrong (it's limit is 0). B is also wrong because it's limit is -1. D is also wrong because its limit as h goes to 0 is 2/x^2, which is close, but no cigar. That leaves either A or C, as both of these would give the correct limit.

We can now do the math.