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Math Help - Derivative questions?

  1. #1
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    Derivative questions?

    I'm really not understanding. I thought 4/3 and pi were constants. So I figured the answer was 2ra+r^2
    But that is wrong.
    Why?
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    Last edited by melliep; September 28th 2010 at 11:00 PM.
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  2. #2
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    4/3 and pi are constants. They are numbers, connected to the variable r by multiplication.

    V(r) = \frac{4}{3}\pi r^2 a

    V(r) = \frac{4\pi r^2 a}{3}

    Now we just differentiate the numerator, ignoring the denominator and then simplify afterwards.

    Can you show us your steps of how you got your answer?
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  3. #3
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    Well, I just took out 4/3 and pi because I figured since they were constants, their derivatives were nothing? But it seems like you did something different.

    Then I thought f = r^2 and g = a
    and then f' = 2r and g' = 1

    And since they're being multiplied I used the product rule. f'g + fg'

    Which gave me (2r)(a) + (r^2)(1)

    But that is apparently incorrect? What am I doing wrong?
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  4. #4
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    No. Differentiation is linear.

    The product rule is only applied if the same variable is being multiplied. Eg: x^2 * cos(x) you would need to use the product rule because it involves the same variable twice.

    Consider the following:

    f(x) = 2x^2 + 3

    f'(x) = 4x + 0

    Correct?
    The derivatives are only 0 if they are seperated by a plus or minus. Otherwise, think of the whole thing as a group.
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  5. #5
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    Oh, then I never learned how to do this.
    But couldn't you take 4/3 and pi out still? Because they are constants regardless?
    So then how do you find the derivative with different variables?
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  6. #6
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    1. You can't take out 4/3 and pi. They are constants, but are apart of r^2. I don't see how you cannot understand this yet you know the product rule...

    Example:

    f(x)=\frac{7}{9}\pi r^2

    f'(x)=2 \cdot \frac{7}{9} \cdot \pi r

    See how they still stay with the variable r and don't get taken out?

    2. With different variables, think of them as another constant, like pi.
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  7. #7
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    I'm sorry, but I never learned this. I'm not quite following.

    [(4)(pi)(r^2)(a)]/3

    Would become [(8)(pi)(r)(a)]/3

    Then what? I really don't understand why you keep the numbers. And I don't know what to do next because everything I thought is wrong apparently.

    If I were to think of them as constants, in my mind and from what I've learned and thought this whole time, you would get rid of them.
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  8. #8
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    You have got the derivative correct!

    You keep them because differentiation is linear.

    The constants are connected by the variable by a multiplication or division, and so they are kept together. Constants only get taken away in differentiation if they are NOT connected to the variable you are differentiating with.
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