Differentiation of the product of a constant and a function

Let $y = cv$. By the General Rule,

• FIRST STEP. $y + \Delta y = c(v + \Delta v) = cv + c\Delta v$.
• SECOND STEP. $\Delta y = c \cdot \Delta v$.
• THIRD STEP. $\frac{\Delta y}{\Delta x} = c\frac{\Delta v}{\Delta x}$.
• FOURTH STEP. $\frac{dy}{dx} = c\frac{dv}{dx}$. [Applying Theorem 3.8.2]

Therefore, $\frac{d}{dx} (cv) = c \frac{dv}{dx}$ (equation (IV) above).

The derivative of the product of a constant and a function is equal to the product of the constant and the derivative of the function.