Find the rate of change of the area of a circle with respect to
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    Application of Derivatives
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    How fast is the volume of a ball changing with respect to its radius when the radius is 3 m?

    Let V be volume of ball of radius r.
     therefore space space space space space space space straight V space equals space 4 over 3 πr cubed
    Rate of change of volume with respect to straight r space equals space dV over dr
                                              equals space straight d over dr open parentheses fraction numerator 4 straight pi over denominator 3 end fraction straight r cubed close parentheses space equals space fraction numerator 4 straight pi over denominator 3 end fraction cross times 3 straight r squared space equals space 4 πr squared
    When r = 3 m, rate of change of volume  = 4 space straight pi space left parenthesis 3 right parenthesis squared space equals space 36 space straight pi space space straight m cubed divided by straight m
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    Find the rate of change of the area of a circle with respect to its radius r when
    (a) r = 3 cm   (b) r = 4 cm

    Let A be area of circle of radius r
    therefore space space space space space space space space space space space space space space space straight A space equals space πr squared
    Rate of change of area with respect to straight r space equals space dA over dr
                                                             equals space straight d over dr left parenthesis πr squared right parenthesis space equals space 2 πr

    (a) When r = 3, rate of change of area = 2straight pi × 3 = 6  cm2/cm.
    (b) When r = 4, rate of change of area = 2 straight pi × 4 = 8 straight pi cm2/cm.

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    Find the rate of change of the area of a circle with respect to its radius r when r = 5 cm.

    Let A be area of circle of radius r
    therefore space space space space space space space space space space space straight A space equals space πr squared
    Rate of change of area with respect to straight r space equals space dA over dr
                                                           equals space straight d over dr left parenthesis πr squared right parenthesis space equals space 2 πr
    when r = 5,    rate of change of area  = 2 straight pi space cross times space 5 space equals space 10 straight pi space cm squared divided by cm
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    Find the rate of change of the volume of a ball with respect to its radius r. How fast is the volume changing with respect to the radius when the radius is 2 m?

    Let V be volume of ball of radius r
    therefore space space space space space space space space space straight V equals space 4 over 3 πr cubed
    Rate of change of volume with respect to straight r space equals space dV over dr
                                    equals space straight d over dr open parentheses 4 over 3 πr cubed close parentheses space equals space 4 over 3 straight pi straight d over dr left parenthesis straight r cubed right parenthesis space equals fraction numerator 4 straight pi over denominator 3 end fraction cross times 3 straight r squared space equals space 4 πr squared
    When r = 2 m, rate of change of volume = 4 straight pi (2)2 = 16 straight pi m3/m.
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    The cost function C(x), in rupees, of producing x items (x ≥ 15) in a certain factory is given by straight C left parenthesis straight x right parenthesis space equals space 20 plus 10 straight x squared plus 15 over straight x.  Determine the marginal cost function and the marginal cost of producing 100 items. 

    Here,   straight C left parenthesis straight x right parenthesis space equals space 20 plus 10 straight x squared plus 15 over straight x space space space space rightwards double arrow space space space space space space straight C space equals space 20 plus 10 straight x squared plus 15 straight x to the power of negative 1 end exponent
    Marginal cost function equals space dC over dx space equals space 0 plus 20 straight x minus 15 straight x to the power of negative 2 end exponent space equals space 20 straight x minus space 15 over straight x squared
    When x = 100,  marginal cost  = 20 (100)  - fraction numerator 15 over denominator left parenthesis 100 right parenthesis squared end fraction
                                                equals space 2000 minus 15 over 10000 space equals space left parenthesis 2000 minus.001 right parenthesis space nearly space equals space Rs. space 2000 space nearly.

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