Derivatives and Integrals Of Sine and Cosine

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Take the Derivative & Integrals Of Sine and Cosine Quickly
Take the Derivatives GRAPHICALLY
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Calculus Lessons Table of Contents
Use a Pencil to Take a Derivative or Antiderivative of Sine & Cosine Functions
Just the Graphs of Sine, Cosine, -Sine, -Cosine for "Taking a derivative Using a Pencil"
Proof that Dxsin(c) = cos(x)

      The Sketch pad material just below is the technologically up-to-date version of the content of this page.

      This page has a "script" and may be used for notes.

      If your browser is Java-able, use the sketches below.

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Visit the Interactive Sketch Pad Material On This Topic

Take A Derivative or Antiderivative or Sine and Cosine

See these instructions in action on this sketchpad: Derivative of the Sine.


1st: Think of the cosine on the horizontal
as in the positive x axis
(because the COSINE is the HORIZONTAL COMPONENT of a vector),
2nd: Think of the symbol sine on the vertical
as in the positive y axis
(because the SINE is the VERTICAL COMPONENT of a vector)
3rd: Think of and place - sin(x) and - cos(x) in the appropriate spots.
To Take a first or second or third or fourth ... DERIVATIVE,
move one or two or three or four ... turns in a CLOCKWISE direction.
To Take a first or second or third or fourth ... ANTIDERIVATIVE,
move one or two or three or four ... turns in a COUNTER-CLOCKWISE direction.

Take First, Second, Third Derivatives of Sine & Cosine Functions GRAPHICALLY

      Use the graphs to takes the derivatives.

      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.


      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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