Which of the following two expressions are equivalent alternatives to the definition of a derivative?

Question

Which of the following two expressions are equivalent alternatives to the definition of a derivative?

A: lim j -> 0 f(x) - f(x-j) / j

B: lim j -> 0 f(x - j/2) - f(x + j/2)/ j

A only

B only

Both A AND B

Neither A NOR B

Answer 1

B would, in the normal case, be the negative of the derivative. (You can create functions where the limit on B exists but there is no derivative. Consider f(0) = 0 f(x) = 1, x≠0. The limit of B at 0 would be 0, but no derivative exists here).

If you aren't sure on a problem like this, try a function with a non-zero derivative and check for j small.

You can show A is true by letting h =-j, noticing that h also has a limit of 0, and this is the standard definition.

The answer is A only, and would be A only even if B had the opposite definition, as we explain above. If you condition the limit definition on cases where the derivative exists, then the negative of B would be true.