I'll work back wards on this one. So I start with:

" alt=" = \lim_{k,h \to 0^{+}} (\frac{1}{h+k})(\;f(x+h) - f(x) + f(x) - f(x-k)\

" />

" alt=" = \lim_{k,h \to 0^{+}} (\frac{h}{h+k})(\;\frac{f(x+h) - f(x)}{h} + \frac{f(x) - f(x-k)}{h}\

" />

" alt=" = \lim_{k,h \to 0^{+}} (\frac{h}{h+k}) \cdot ( \; \lim_{k,h \to 0^{+}}\frac{f(x+h) - f(x)}{h} + \lim_{k,h \to 0^{+}}\frac{f(x) - f(x-k)}{h}\

" />

" alt=" = \lim_{k,h \to 0^{+}} (\frac{h}{h+k}) \cdot ( \; f'(x) + \lim_{k,h \to 0^{+}}\frac{f(x) - f(x-k)}{h}\

" />

" alt=" = \lim_{k,h \to 0^{+}} (\frac{1}{h+k}) \cdot ( \; f'(x) \cdot (\lim_{k,h \to 0^{+}} \; h) + \lim_{k,h \to 0^{+}}f(x) - f(x-k)\

" />

" alt=" = \lim_{k,h \to 0^{+}} (\frac{k}{h+k}) \cdot ( \; \frac{1}{k}f'(x) \cdot (\lim_{k,h \to 0^{+}} \; h) + \lim_{k,h \to 0^{+}}\frac{f(x) - f(x-k)}{k}\

" />

\cdot f'(x) \cdot (\lim_{k,h \to 0^{+}} \; h) + f'(x)" alt=" = \lim_{k,h \to 0^{+}} (\frac{k}{h+k}) \cdot ( \; ( \;\lim_{k,h \to 0^{+}}\frac{1}{k} \

\cdot f'(x) \cdot (\lim_{k,h \to 0^{+}} \; h) + f'(x)" />

\cdot f'(x)" alt=" = [\lim_{k,h \to 0^{+}} \frac{1}{h+k}] \cdot ( \;\;\lim_{k,h \to 0^{+}} h+ k \;\

\cdot f'(x)" />