What is the geometric interpretation of the constant multiple rule $(cf)' = cf'$?

Conditions

• $c$ is a constant.
• The function $f$ is differentiable.

Reasoning, step by step

1. Start with the product rule: $(cf)' = c'f + cf'$.
2. Recognize that the derivative of a constant $c$ is $c'=0$.
3. Substitute $c'=0$ into the equation.
4. Conclude that $(cf)' = 0 \cdot f + c f' = c f'$.
5. Visualize this as scaling the vertical heights of the graph by $c$, which scales the slope by $c$.

Example

If $f(x)=x^2$, then $(3x^2)' = 3(2x) = 6x$. The graph is stretched vertically by 3, so the slope at any point is 3 times steeper.

Common misconceptions

• Thinking that the constant affects the input variable $x$ rather than the output $f(x)$.
• Believing that the product rule does not apply to constants.
• Confusing constant multiplication with function composition.

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Explore next

• How does the product rule reduce to the constant multiple rule?
• What is the derivative of a constant function?
• Why is feeding one function into another different from multiplying their outputs?