calculus

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find derivative of (8sqrt(x)+(9/2root3(x)))^2

  • parentheses required -

    You have not supplied sufficient parentheses to render the expression unambiguous, that probably explains why you get different results from different sources.

    Is it
    (9/2)root3(x)
    or
    9/(2root3(x)) ?

  • calculus -

    its like 9/2(x)^1/3

  • calculus -

    It is still not clear to me if (x)^1/3 is with the numerator or the denominator.

    I assume you are transcribing from a type-set fraction where the paretheses around the denominator are understood. When transcribing to a single line (as in this case), you will need to insert explicitely the parentheses around ALL denominators and numerators to avoid ambiguity.

  • calculus -

    its in denominator

  • calculus -

    So we're looking at find the derivative of the following expression:
    y=(8*sqrt(x)+((9/2)/(x)^(1/3)))^2

    Using the chain rule, we get
    dy/dx
    = 2(8*sqrt(x)+((9/2)/(x)^(1/3))) * d(8*sqrt(x)+((9/2)/(x)^(1/3)))/dx

    By writing
    =(8*sqrt(x)+((9/2)/(x)^(1/3)))^2
    =8x^(1/2) + (9/2)x^(-1/3)
    We can differentiate u using the power rule:
    du/dx = 8(1/2)x^(-1/2) + (9/2)(-1/3)x^(-4/3)

    So dy/dx
    =2(8*sqrt(x)+((9/2)/(x)^(1/3)))*(8(1/2)x^(-1/2) + (9/2)(-1/3)x^(-4/3) )
    =64+12x^(5/6)-(27/2)x^(-5/3)
    after simplification.
    Check my work.

  • calculus -

    is this the final answer?

  • calculus -

    Yes it is, but you should check my work to make sure you understand how this is done, and that I did not make a mistake.

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