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Mathematics LibreTexts

Derivatives The Easy Way

  • Page ID
    623
  • [ "article:topic" ]

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    Constant Rule and Power Rule

    We have seen the following derivatives:

    1. If f(x) = c, then f '(x) = 0
    2. If f(x) = x, then f '(x) = 1
    3. If f(x) = x2, then f '(x) = 2x
    4. If f(x) = x3, then f '(x) = 3x2
    5. Iff(x) = x4, thenf '(x) = 4x3

    This leads us the guess the following theorem.
     

    Theorem

          d
                xn  =  nxn-1
         dx 


    Proof:

    We have

           


    Applications

    Example

    Find the derivatives of the following functions:

    1. f(x) = 4x3 - 2x100

    2. f(x) = 3x5 + 4x8 - x + 2

    3.  f(x) = (x3 - 2)2

    Solution  

    We use our new derivative rules to find

    1. 12x2 - 200x99

    2. 15x3+32x7-1

    3. First we FOIL to get

              [x6 - 4x3 + 4] ' 

      Now use the derivative rule for powers

              6x5 - 12x2


    Example:

    Find the equation to the tangent line to 

            y  =  3x3 - x + 4 

    at the point(1,6)

    Solution:

            y'  =  9x2 - 1 

    at x = 1 this is 8. Using the point-slope equation for the line gives

            y - 6  =  8(x - 1) 

    or 

            y  =  8x - 2


    Example:

    Find the points where the tangent line to 

            y  =  x3 - 3x- 24x + 3 

    is horizontal.

    Solution:

    We find 

            y'  =  3x2 - 6x - 24

    The tangent line will be horizontal when its slope is zero, that is, the derivative is zero.  Setting the derivative equal to zero gives:

            3x2 - 6x - 24  =  0 

    or

            x2 - 2x - 8  =  0 

    or

            (x - 4)(x + 2)  =  0

    so that 

            x = 4    or    x = -2
     


    Derivative of f(x) = sin(x)

     

    Theorem

          d
                sin(x)  =  cos(x)
         dx           

     


    Proof:  




     


    d/dx cos(x)

     

    Theorem

           d   
                 cos x  =  -sin x
          dx

     

    Larry Green (Lake Tahoe Community College)