Derivative rules via approximations
The fact that the derivative [math]\displaystyle{ f'(a) }[/math] exists can be written as [math]\displaystyle{ f(x) \approx f(a) + f'(a) (x-a) }[/math] for all [math]\displaystyle{ x }[/math] that are near [math]\displaystyle{ a }[/math]. This is known as linear approximation (or linearization) of f at the number a. Various rules for differentiation can be derived using linear approximation.
Constant multiple rule. Let [math]\displaystyle{ h(x) = k f(x) }[/math], where [math]\displaystyle{ k }[/math] is a constant. Then for all [math]\displaystyle{ x }[/math] near [math]\displaystyle{ a }[/math] we have [math]\displaystyle{ g(x) = kf(x) \approx k \left[ f(a) + f'(a) (x-a) \right] = kf(a) + kf'(a) (x-a). }[/math] Since [math]\displaystyle{ g(a) = kf(a) }[/math], we obtain [math]\displaystyle{ g(x)\approx g(a) + {\color{red} kf'(a)} (x-a) }[/math]. Comparing it with [math]\displaystyle{ g(x)\approx g(a) + {\color{red} g'(a)} (x-a) }[/math] we conclude that [math]\displaystyle{ g'(a) = k f'(a) }[/math], or (replacing [math]\displaystyle{ a }[/math] with an arbitrary variable):
[math]\displaystyle{ \boxed{(kf(x))' = kf'(x)} }[/math]
Sum/Difference rule. Let [math]\displaystyle{ h(x) = f(x) + g(x) }[/math]. For all [math]\displaystyle{ x }[/math] near [math]\displaystyle{ a }[/math] we have
[math]\displaystyle{ h(x) = f(x) + g(x)\approx \left[ f(a) + f'(a) (x-a) \right] + \left[ g(a) + g'(a) (x-a) \right] = \left[f(a) +g(a)\right]+ \left[f'(a) +g'(a)\right](x-a) }[/math].
Since [math]\displaystyle{ h(a) = f(a) +g(a) }[/math], we obtain [math]\displaystyle{ h(x)\approx h(a) + {\color{red}\left[ f'(a)+g'(a)\right]} (x-a) }[/math]. Comparing it with [math]\displaystyle{ h(x)\approx h(a) + {\color{red} h'(a)} (x-a) }[/math] we conclude that [math]\displaystyle{ h'(a) = f'(a)+g'(a) }[/math], or (replacing [math]\displaystyle{ a }[/math] with an arbitrary variable):
[math]\displaystyle{ \boxed{(f(x)+g(x))' = f'(x)+g'(x)} }[/math]
Product rule. Let [math]\displaystyle{ h(x) = f(x)g(x) }[/math]. For all [math]\displaystyle{ x }[/math] near [math]\displaystyle{ a }[/math] we have
[math]\displaystyle{ h(x) = f(x) g(x)\approx \left[ f(a) + f'(a) (x-a) \right] \cdot \left[ g(a) + g'(a) (x-a) \right] = f(a)\cdot g(a)+ {\color{red} \left[f'(a)g(a) +f(a)g'(a)\right]}(x-a) + {\color{green} f'(a)g'(a)(x-a)^2}\ }[/math].
Since [math]\displaystyle{ h(a) = f(a)g(a) }[/math], we obtain [math]\displaystyle{ h(x)\approx h(a) + {\color{red}\left[ f'(a)g(a)+f(a)g'(a)\right]} (x-a) }[/math]. Comparing it with [math]\displaystyle{ h(x)\approx h(a) + {\color{red} h'(a)} (x-a) }[/math] we conclude that [math]\displaystyle{ h'(a) = f'(a)g(a)+f(a)g'(a) }[/math], or (replacing [math]\displaystyle{ a }[/math] with an arbitrary variable):
[math]\displaystyle{ \boxed{(f(x)g(x))' = f'(x)g(x)+f(x)g'(x)} }[/math]