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 D_xsin(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.

Visit the Interactive Sketch Pad Material On This Topic Derivative of the Sine   -- full blown 4-page sketches Derivative of the Cosine -- take the derivative with a tangent line, with trace

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.

      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.

      For example, TAKE THE DERIVATIVE OF THE SINE.

• 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.
• 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.

      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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