Taylor Expansion of Composition of Functions

Question

I have in my lecture notes that $$f(g(x+h))-f(g(x))\approx f'(g(x))\left(g(x+h)-g(x)\right)+f''(g(x))(g(x+h)-g(x))^2$$

He explained can found via taylor expansion, but I try to expand it and am not getting this answer

Answer 1

For brevity in notation, let $y := g(x)$ and $k:= \def\k{g(x+h) - g(x)}\k$. Then, by Taylor's formula, \begin{align*} & f\bigl(g(x+h)\bigr) - f\bigl(g(x)\bigr) \\&= f(y+k) - f(y)\\ &= f'(y)k + \frac 12 f''(y)k^2 + o(k^2)\\ &= f'\bigl(g(x)\bigr)\bigl(\k\bigr) + \frac12 f''\bigl(g(x)\bigr)\bigl(\k\bigr)^2 + o\Bigl(\bigl(\k\bigr)^2\Bigr) \end{align*} So you missed a $\frac 12$.