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State each differentiation rule both in symbols and words.

(a) The Power Rule

(b) The Constant Multiple Rule

(c) The Sum Rule

(d) The Difference Rule

(e) The Product Rule

(f) The Quotient Rule

(g) The Chain Rule

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Angelo Torreno
Angelo TorrenoLv4
4 Nov 2020

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Related questions

#1 & 2

How do you find the integral of ?

To solve: The integral of 

Formulas to be used: (in order of appearance in the Solution part below)

  • Trigonometric double-angle identity: 
  • Difference rule of integration:   
  • Constant multiple rule of integration:   
  • Constant multiple rule of differentiation: 
  • Trigonometric rule of integration: 

Solution:

The given function  has no direct integral, but can be solved by thinking of a trigonometric relationship that can be easily manipulated and integrated. In this case, the double-angle identity for cosine is used.  

                Using the double-angle identitiy for cosine

                Isolating the  term on the left side

             Making the coefficient of  term to be 1

The working equation for  can now be integrated using -substitution, as will be shown.

    Integrating both sides of the equation

                      Taking the integral using the difference rule  and

                                                          constant multiple rule 

 and                          Using -substitution and differentiating it using the constant multiple rule 

                  Replacing the values of  and 

                    Taking the integral using the trigonometric rule 

                  Substituting back the value of , which is 

Answer:

   

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