How Derivatives Show a Rate of Change
Differentiation is the process of finding derivatives. The derivative of a function tells you how fast the output variable (like y) is changing compared to the input variable (like x).
For example, if y is increasing 3 times as fast as x — like with the line y = 3x + 5 — then you say that the derivative of y with respect to x equals 3, and you write
This, of course, is the same as
and that means nothing more than saying that the rate of change of y compared to x is in a 3-to-1 ratio, or that the line has a slope
The following practice questions emphasize the fact that a derivative is basically just a rate or a slope. So to solve these problems, all you have to do is answer the questions as if they had asked you to determine a rate or a slope instead of a derivative.
If you leave your home at time = 0, and speed away in your car at 60 miles per hour, what’s
the derivative of your position with respect to time?
What’s the slope of the parabola
at the point (7, 9)? (Refer to the following figure.)
Answers and explanations
The answer is
A derivative is always a rate, and (assuming you’re talking about instantaneous rates, not average rates) a rate is always a derivative. So, if your speed, or rate, is
is also 60.
The slope is 3.
You can see that the line, y = 3x – 12, is tangent to the parabola,
at the point (7, 9). You know from y = mx + b that the slope of y = 3x – 12 is 3. At the point (7, 9), the parabola is exactly as steep as the line, so the derivative (that’s the slope) of the parabola at (7, 9) is also 3.
Although the slope of the line stays constant, the slope of the parabola changes as you climb up from (7, 9), getting less and less steep. Even if you go to the right just 0.001 to x = 7.001, the slope will no longer be exactly 3.