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

Derivatives The derivative of:

a CONSTANT
is
zero.

the PRODUCT OF A CONSTANT AND A FUNCTION

is
the product of the constant and the derivative of the function.


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

is
the derivative of the first times the second plus the derivative of the second times the first.


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

is
the derivative of the first times the second minus the derivative of the second times the first, ALL
divided by the square of the second.


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

is
the product of the derivative of the outer evaluated at the inner and the derivative of the inner.

Arithmetic Tricks, Rules & Shortcuts
In 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.

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.

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.


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…


Written by Sahil K . A

It is not true that people stop pursuing dreams because they grow old, they grow old because they stop pursuing dreams.
– Gabriel Garcia Marquez

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