Use a Pencil to Take Derivatives of Sine & Cosine Functions


      The derivative is a slope. You will be using a pencil as a tangent to a curve. As you move the tangent from the left to the right along the curve, you will also describe in words the slope of that tangent which is also the slope of the curve at that point. So, as you move the tangent pencil and state the slopes aloud, you are taking the derivative of the curve.

      This page permits the user to choose between a hands-on, pencil & paper use, and a digital presentation.

sketchpad sketchpad
Visit the Interactive Sketch Pad Material On This Topic
  • DerCos   -- cosine graph, movable tangent line w/equation,
        * m derived by simple m formula (h=.01),
        * "derivative" may be graphed.
  • DerivSin   -- full blown 4-page sketches ** Use this
  • SinTraceDerSums -- Sine from many views, right triangle, ABCD, derivative, antiderivative


    1. Sharpened Pencil

    2. One of these pictures of graphs of the trig functions:


First Derivative

      To take the first derivative, use a pencil. Use the middle point of a pencil as a tangent point and point the pencil to the right, the greater x values. Trace the curve, stating the derivative (slope of the tangent) as you do.

      Use the stated derivatives (slopes) to describe the curve which is the derivative functions.


  • At zero the curve is increasing at a 45% angle so, the slope is 1.
  • In the first quadrant, the function is increasing but at a slower rate, the slope is decreasing but positive.
  • At 90°, the curve reaches a relative max, the slope is 0.
  • In the second quadrant, the function is still decreasing, the derivative (slope) is negative.
  • At 180°, the function is at a -45° angle, the slope is -1.
  • In the third quadrant, the curve continues to decrease, but not as sharply, the slope is negative but increasing.
  • At 270°, the function reaches a relative min, and the slope is 0.
  • In the third quadrant, the function increases, the derivative (slope) is positive.
  • At 360°, the function still increasing, the slope is 1 and the cycle begins to repeat.

      What's the function?

      The cosine.

      The derivative of the sine is the cosine.


Second Derivative

      Two methods for taking the second derivative, the slope of the derivative, are suggested. EITHER repeat the above method using the cosine as the original function, OR, use the movement of the PENCIL POINT and the sine function to compute the second derivative of the sine, d2[sin(x)]/dx2.

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