How to Find the Derivative of a Curve

Calculus is the mathematics of change — so you need to know how to find the derivative of a parabola, which is a curve with a constantly changing slope.

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The figure below shows the graph of the above parabola.

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Notice how the parabola gets steeper and steeper as you go to the right. You can see from the graph that at the point (2, 1), the slope is 1; at (4, 4), the slope is 2; at (6, 9), the slope is 3, and so on.

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x (horizontal position) 1 2 3 4 5 6 etc.
y (height) 0.25 1 2.25 4 6.25 9 etc.
1/2 x (slope) 0.5 1 1.5 2 2.5 3 etc.

Here’s the calculus. You write

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And you say,

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Or you can say,

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  1. Take the power and put it in front of the coefficient.

    image6.png
  2. Multiply.

    image7.png

    (Note that this is only a temporary, interim result on the road to the solution below; by itself, it is meaningless.)

  3. Reduce the power by 1.

    In this example, the 2 becomes a 1. So the derivative is

    image8.png
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