Proof of Various Derivative Properties — Question 2

Question 2

Assume that $f$ is differentiable at $x=a$ and that $c$ is a constant. Prove that $$(cf)'(a)=c\,f'(a).$$

Question 2 - Solution

By definition of the derivative, $$(cf)'(a)=\lim_{h\to 0}\frac{cf(a+h)-cf(a)}{h}.$$

Factor out the constant $c$ from the numerator: $$(cf)'(a) = \lim_{h\to 0}\frac{c\bigl(f(a+h)-f(a)\bigr)}{h}.$$

Rewrite the fraction: $$(cf)'(a) = c\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}.$$

Since $f$ is differentiable at $a$, the limit exists and equals $f'(a)$. Therefore, $$(cf)'(a)=c\,f'(a).$$