Basic Rules + Proofs

Constant Rule

Statement:

If $f(x) = c$ where $ c $ is a constant, then $f'(x) = 0$.

Proof:

Using the limit definition of a derivative, $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ $f'(x) = \lim_{h \to 0} \frac{c - c}{h}$ $f'(x) = \lim_{h \to 0} 0 = 0$

Sum Rule

Statement:

If $f(x)$ and $g(x)$ are differentiable functions, then the derivative of $f(x) + g(x)$ is $f'(x) + g'(x)$.

Proof:

$\frac{d}{dx} [f(x) + g(x)] = \lim_{h \to 0} \frac{f(x+h) + g(x+h) - f(x) - g(x)}{h}$ $= \lim_{h \to 0} \left[ \frac{f(x+h) - f(x)}{h} + \frac{g(x+h) - g(x)}{h} \right]$ $= f'(x) + g'(x)$

Constant Multiple Rule

Statement:

If $f(x)$ is a differentiable function and $ c $ is a constant, then the derivative of $cf(x)$ is $cf'(x)$.

Proof:

$\frac{d}{dx} [cf(x)] = \lim_{h \to 0} \frac{c[f(x+h) - f(x)]}{h}$ $= c \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ $= cf'(x)$

Of course, here's the proof with the replacements made: