Coefficient rule in calculus

Statement

The derivative of a function with a coefficient is the coefficient times the derivative of that function.

$$\tfrac{d}{dx}\bigg(c*g(x)\bigg) = c * g'(x)$$

Proof

Define function $f$.

$$f(x) = c * g(x)$$

Take the definition of the derivative.

$$f'(x) = \lim_{h \to 0}\left(\frac{f(x + h) - f(x)}{h}\right)$$

$$f'(x) = \lim_{h \to 0}\left(\frac{c * g(x + h) - c * g(x)}{h}\right)$$

Factor out the $c$ and move it out the fraction.

$$f'(x) = \lim_{h \to 0}\left(\frac{c * \bigg(g(x + h) - g(x)\bigg)}{h}\right)$$

$$f'(x) = \lim_{h \to 0}\left( c * \frac{g(x + h) - g(x)}{h}\right)$$

Move the $c$ out the limit.

$$f'(x) = c * \lim_{h \to 0}\left(\frac{g(x + h) - g(x)}{h}\right)$$

Now the definition of the derivative of $g(x)$ arised.

$$f'(x) = c * g'(x)$$