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**Derivatives and Integration’s Arithmetic Tricks, Rules & Shortcuts**

**Derivatives** The derivative of:

a CONSTANT is zero. | |

the PRODUCT OF A CONSTANT AND A FUNCTION is | |

the PRODUCT OF TWO FUNCTIONS, a first and a second function, is | |

the QUOTIENT OF TWO FUNCTIONS, a first divided by a second function, is | |

a FUNCTION OF A FUNCTION, a major function and its argument function, an outer function and the inner function, is |

Arithmetic Tricks, Rules & ShortcutsIn Words & Symbols |

- Factor-Out A Constant
- Simplify Through Symmetry
- Switch the Limits and Get the Opposite
- Rewrite Using Zero As A Limit, If Possible
- Rewrite as A Sum By Seperating Terms
- Rewrite as A Sum By Using More Limits of Integration

**The sine and cosine are closely related. **

- One is the derivative or opposite of the derivative of the other.
- One is the antiderivative or the opposite of the antiderivative of the other.

Take A Derivative or Antiderivative or Sine and Cosine |

- 1st: Place the symbol cosine on the horizontal
- as in the positive x axis
- (as in the cosine is the horizontal component of a vector),
- 2nd: Place the symbol sine on the vertical
- as in the positive y axis
- (as in the sine is the vertical component of a vector)
- 3rd: 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.

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.

The derivative of the sine is the cosine.

**To integrate a function times the derivative of another function, use:**

================================================ Imp Note: IF u have Any Doubt pls Don’t ask me.. i hate this subject…

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