> [!summary] This proof uses the definition of a limit to prove the constant multiple rule. > **Key Result:** $\frac{d}{dx}[C\cdot F(x)] = C\cdot F'(x)$ >[!info]+ Read Time **⏱ 1 min** # Mathematical Proof > [!warning] Assumptions To prove the derivative of a constant and a function using [[Direct Proof|direct proof]] assume the following: > - The definition of a [[Derivative|derivative]] is $\frac{dy}{dx} = f'(x) = \displaystyle \lim_{ h \to 0 } \frac{f(x+h)-f(x)}{h}$ Prove that $\frac{d}{dx}[C\cdot F(x)] = C\cdot F'(x)$ $ \begin{align*} \frac{d}{dx}[C\cdot F(x)] &= \displaystyle \lim_{ h \to 0 } \frac{C \cdot f(x+h)- C\cdot f(x)}{h} \\ &=\displaystyle C \cdot\lim_{ h \to 0 } \frac{ f(x+h)- \cdot f(x)}{h} \\ &= C \cdot F'(x) \end{align*} $ --- > 📚 Like this note? [Star the GitHub repo](https://github.com/rajeevphysics/Obsidian-MathMatter) to support the project and help others discover it! ---