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Math Help - Derivative Help

  1. #1
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    Derivative Help

    Hey, I am running through some examples trying to get ready for my test and am completely stuck on one. I have a feeling something similar will be on our test too!

    Anyways here is the question


    Using the definition of the derivative find f ' (x). Then find f ' (-2), f ' (0), and f ' (3).

    Problem: f(x) = -3 square root of x

    Sorry, I can;t figure out how to make the square root sybol but you get the idea..-3 is on the outside with x inside the square root symbol. I mainly need to know f ' (x) and then obviously finding the others will be pretty easy. Thanks in advance!
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  2. #2
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    Quote Originally Posted by Mathamateur View Post
    Hey, I am running through some examples trying to get ready for my test and am completely stuck on one. I have a feeling something similar will be on our test too!

    Anyways here is the question


    Using the definition of the derivative find f ' (x). Then find f ' (-2), f ' (0), and f ' (3).

    Problem: f(x) = -3 square root of x

    Sorry, I can;t figure out how to make the square root sybol but you get the idea..-3 is on the outside with x inside the square root symbol. I mainly need to know f ' (x) and then obviously finding the others will be pretty easy. Thanks in advance!
    Been a while since I did one of these!

    f(x) = -3\sqrt{x}

    f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

    So:
    f'(x) = \lim_{h \to 0} \frac{-3\sqrt{x+h} + 3\sqrt{x}}{h}

    We need to rationalize the numerator:
    f'(x) = \lim_{h \to 0} \frac{-3\sqrt{x+h} + 3\sqrt{x}}{h} \cdot \frac{-3\sqrt{x+h} - 3\sqrt{x}}{-3\sqrt{x+h} - 3\sqrt{x}}

    f'(x) = \lim_{h \to 0} \frac{9(x + h) - 9x}{h(-3\sqrt{x+h} - 3\sqrt{x})}

    f'(x) = \lim_{h \to 0} \frac{9x + 9h - 9x}{h(-3\sqrt{x+h} - 3\sqrt{x})}

    f'(x) = \lim_{h \to 0} \frac{9h}{h(-3\sqrt{x+h} - 3\sqrt{x})} =  \lim_{h \to 0} \frac{9}{-3\sqrt{x+h} - 3\sqrt{x}}

    Now take the limit:
    f'(x) = \frac{9}{-3\sqrt{x} - 3\sqrt{x}} = -\frac{9}{2 \cdot 3\sqrt{x}} = -\frac{3}{2\sqrt{x}}

    I'll let you take it from here.

    -Dan

    PS Why are f'(-2) and f'(0) undefined? Describe what's happening to the function at f(0).
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