<s>[INST] Error in finding the derivative of the general quadratic function $f(x)=ax^2+bx+c$ Where is the error in this 'proof' that the derivative of the quadratic function is equal to $x(2a+1)$? Note: I use $h$ to denote a small number.
$$f(x)=ax^2+bx+c\\f'(x)=\lim\limits_{h \to 0} \frac{a(x+h)^2+b(x+h)+c-(ax^2+bx+c)}{h}\\=\lim\limits_{h \to 0}\frac{ax^2+2ahx+ah^2+bx+hx+c-ax^2-bx-c}{h}\\=\lim\limits_{h \to 0}\frac{2ahx+ah^2+hx}{h}\\=\lim\limits_{h \to 0}2ax+ah+x\\=2ax+x\\=x(2a+1)$$
 [/INST] You wrote  $$\lim\limits_{h \to 0}\frac{ax^2+2ahx+ah^2+bx+\color{red}{hx}+c-ax^2-bx-c}{h}$$ instead of $$\lim\limits_{h \to 0}\frac{ax^2+2ahx+ah^2+bx+\color{blue}{bh}+c-ax^2-bx-c}{h}.$$
Simplifying, it becomes
$$\lim\limits_{h \to 0}\frac{ax^2+2ahx+ah^2+bx+bh+c-ax^2-bx-c}{h}=\lim_{h\to0}\frac{h(2ax+b)}{h}=2ax+b$$
as desired.
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