Question

Use the Product Rule to show that \(\frac{d}{d x}(c \cdot f(x))=c \cdot \frac{d}{d x} f(x)\) for any constant c.

Short answer

The derivative of a constant multiplied by a function equals the constant times the derivative of the function. This can be confirmed by applying the definition of the derivative and factoring out the constant c.

Step-by-step solution

Understand the Problem

We are asked to prove that the derivative of a function multiplied by a constant is equal to the constant multiplied by the derivative of the function. This rule is crucial in differentiating many elementary functions.

Start proof using the Definition of Derivative

We first write down the definition of derivative for the function \(c \cdot f(x)\) as the limit: \[\frac{d}{d x}(c · f(x)) = \lim_{h \rightarrow 0} \frac{c \cdot f(x + h) - c \cdot f(x)}{h}\]

Simplify the expression

Factor out the constant c from the numerator to simplify the limit expression: \[\lim_{h \rightarrow 0} \frac{c \cdot f(x + h) - c \cdot f(x)}{h} = c \cdot \lim_{h \rightarrow 0} \frac{f(x + h) - f(x)}{h}\]

Recognize the Derivative

Notice that the limit now is the definition of the derivative of the function \(f(x)\): \[c \cdot \lim_{h \rightarrow 0} \frac{f(x + h) - f(x)}{h} = c \cdot \frac{d}{d x} f(x)\] Thus, we have shown that \(\frac{d}{d x}(c \cdot f(x)) = c \cdot \frac{d}{d x} f(x)\) thus confirming the constant multiple rule for derivatives.