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 maor function and its argument function, an outer function and the inner function, is
   

Arithmetic Tricks, Rules & Shortcuts

• 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

Arithmetic Tricks, Rules & Shortcuts

• 
   
• 
   
• 
   
• 
   
• , where f(x) is an even funtion
   
• 
   
• 
   

Arithmetic Tricks, Rules & Shortcuts

• 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.

  Derivative & Antiderivative of Sine & Cosine Functions

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

     

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