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Mathematics LibreTexts

Derivatives The Easy Way

  • Page ID
    623
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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 - 3x2 - 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)