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Math Help - How do I differentiate this?

  1. #1
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    How do I differentiate this?

    f(x) = 3/quarticroot(9x^2+4)

    quarticroot = the square root symbol but with a small 4 on the left side of the root.

    Thanks in advance.
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  2. #2
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    Think of the quartic root as, instead, being a 1/4 power (square roots are 1/2 power, cube roots are 1/3 power, etc.) So now your expression becomes:

    3/(9x^2+4)^(1/4).

    You can move your denominator into the numerator simply by making the exponent negative, giving:

    3*(9x^2+4)^(-1/4).

    Now, you can differentiate this using the power rule in combination with the chain rule. Bring the power down and reduce it by 1 - this makes your new exponent (-5/4). Also, the chain rule says you have to take the derivative of "the inside function," here the 9x^2+4 portion.

    This gives 3*(-1/4)(9x^2+4)^(-5/4)*(18x).

    Reading left to right, the 3 is the original constant you had;
    the (-1/4) is the power you brought down via the power rule;
    the (9x^2+4)^(-5/4) is basically your original fourth-root, put in the numerator, with the power reduced by 1;
    the 18x is the derivative of 9x^2+4, present because of the chain rule.

    You can simplify a little bit, of course, but that takes care of the differentiation.

    Hope this helps!
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  3. #3
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    Quote Originally Posted by Jeavus View Post
    f(x) = 3/quarticroot(9x^2+4)

    quarticroot = the square root symbol but with a small 4 on the left side of the root.

    Thanks in advance.
    f(x)=\frac{3}{^4\sqrt{9x^2+4}} \Rightarrow 3(9x^2+4)^{\frac{-1}{4}}

    Take the derivative using power and chain rules...

    f'(x)=\frac{3}{4}(9x^2+4)^{\frac{-5}{4}}(18x)
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