Differentiating Products
Differentiating Powers
UBC Calculus Online Course Notes

Differentiating Constant Multipliers

If we have a function which is multiplied by a constant $ c, $ we will see that its derivative is found by the



Constant Multiplier Rule

\[ \frac{d}{dx}(cy) = c\frac{dy}{dx} \]



This follows easily from the product rule:

\[ \frac{d}{dx}(cy) = \frac{dc}{dx} y + c\frac{dx}{dx} = c\frac{dy}{dx} \]

since the derivative of the constant $ \frac{dc}{dx} = 0. $



Example:

Suppose that $ y(x) = 3x -2. $ What we have seen tells us that

\[ \frac{dy}{dx} = 3\frac{dx}{dx} + \frac{d(-2)}{dx}= 3. \]


This makes sense since the graph of $ y $ is a straight line with slope 3.



Questions:

  1. Can you explain the relationship between these two graphs?

  2. What can you infer from the fact that the derivative is always positive?

  3. What can you infer from the fact that the derivative is constant?


To understand the constant multiplier rule more generally, remember that multiplying a function by a constant $ c $ either stretches or compresses the graph in the vertical direction; that is, if $ c > 1, $ the graph is streched by a factor of $ c. $ This reflects the fact that the rate of change of the function has been increased by a factor of $ c. $