This might seem like straightforward question, but a couple of of us are having actually a conflict over it. We are looking at the following limit below, where \$ f \$ is a real-valued role on an open subset \$ U \$ of \$ londonchinatown.orgbbR \$ that is differentiable at the suggest \$ x_0 \$, and where \$ a \$ and also \$ b \$ are in \$ londonchinatown.orgbbR \$:\$\$lim_h o 0 fracf(x + a h) - f(x + b h)h.\$\$

My friend thinks the this is equal to \$ f"(x_0) \$, but I nothing see how to prove the this is true.

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Thanks for the help!  The easiest method to prove your friend wrong is to use this come a function that we really understand, choose \$f(x) = x\$. We"ll execute it at \$x_0 = 1\$. Then you room calculating

\$\$ fracf(1 + ah) - f(1 + bh)h = frac1 + ah - 1 - bhh = (a - b).\$\$

This is no \$1\$ and also depends critically top top \$a\$ and \$b\$ (exactly choose derivatives perform not).

More generally, we have the right to write (but where I omit all limit signs)\$\$ eginalignfracf(x_0 + ah) - f(x_0 + bh)h &= fracf(x_0 + ah) - f(x_0)h - fracf(x_0 + bh) - f(x_0)h \& o af"(x_0) - bf"(x_0) = (a-b)f"(x_0).endalign\$\$ Thanks because that contributing an answer to londonchinatown.org Stack Exchange!

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