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Math Help - Differentiate (x^2+3)^4

  1. #1
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    Differentiate (x^2+3)^4

    ok i know the answer is 4(x^2+3)^3(2x)

    but i don't understand where did the (2x) come from

    Why wouldn't it just be 4(x^2+3)^3 since the rule is d/dx X^n = nx^(n-1)
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  2. #2
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    Quote Originally Posted by jkami View Post
    ok i know the answer is 4(x^2+3)^3(2x)

    but i don't understand where did the (2x) come from

    Why wouldn't it just be 4(x^2+3)^3 since the rule is d/dx X^n = nx^(n-1)

    Your rule is correct but you must also use the chain rule since your function is a function of a function!

    Let's imagine that  g(x) = x^2+3 , and  f(x) = 4x^{3}

    Then your function can be written  h(x) = f(g(x))

    And the chain rule says that  h'(x) = f'(g(x)) \times g'(x)
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  3. #3
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    Which is the correct answer

    thx
    Last edited by jkami; February 14th 2009 at 05:12 PM. Reason: ...
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  4. #4
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    Which one is correct?

    differentiate

    (x^3+9)^4

    I got two answers from different sources

    answer1 = 12x^2(x^3+9)^3

    answer2 = 12x^3(x^3+9)^3
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  5. #5
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    The first is correct.

    The Chain Rule states, \frac{d}{dx}f(g(x))=f'(g(x))g'(x). The rule works because the rate of change of g(x) multiplies the rate of change of f(g(x)). Hence, we have

    \frac{d}{dx}(x^3+9)^4 = 4(x^3+9)^3 \cdot 3x^2= 12x^2(x^3+9)^3.
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  6. #6
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    Quote Originally Posted by jkami View Post
    differentiate

    (x^3+9)^4

    I got two answers from different sources

    answer1 = 12x^2(x^3+9)^3

    answer2 = 12x^3(x^3+9)^3
    Perhaps an unorthodox way of differentiating but,one ,does not have to do repeated substitutions is the following:


    \frac{d}{dx}(x^3+9)^4=\frac{d(x^3+9)^4}{d(x^3+9)}.  \frac{d(x^3+9)}{dx}=4(x^3+9)^3.3x^2=12x^2(x^3+9)^3
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