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Math Help - Several questions I didn't get

  1. #1
    Junior Member NAPA55's Avatar
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    A couple questions I didn't get

    I did several pages of calculus for homework, and these are the questions I got stuck on:

    1. Find the derivative of y=2/√x + x/√3 + 6(x)^1/3.

    2. Find the derivative at x= -2 for y=3u+2u and u = √(x+5).

    3. The amount of pollution in a certain lake is P(t) = (t^0.25 + 3), where t is years and P is parts per million. What is the rate of change after 16 years (I know I use the limit formula but the exponents and t+h are confusing me).

    4. Find the x-intercept of the function f(x)= 2x^(5/3) - 5x^(2/3).

    5. If f is a differentiable function, find an expression for the derivative of h(x)=2xf(x).

    Thanks!
    Last edited by NAPA55; March 6th 2008 at 12:33 PM.
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  2. #2
    GAMMA Mathematics
    colby2152's Avatar
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    Quote Originally Posted by NAPA55 View Post
    1. Find the derivative of y=2/√x + x/√3 + 6(x)^1/3.
    Do you know how to apply the power rule for derivatives?

    f(x)=ax^n
    f'(x)=nax^{n-1}

    f(x)=\frac{2}{\sqrt{x}} + \frac{x}{\sqrt{3}} + 6x^{\frac{1}{3}}

    Convert all you radicals to exponents

    f(x)=2x^{\frac{1}{2}} + \frac{x}{\sqrt{3}} + 6x^{\frac{1}{3}}

    Now, I will leave the derivation up to you!
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  3. #3
    Senior Member Peritus's Avatar
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    2. Find the derivative at x= -2 for y=3u+2u and u = √(x+5).
    using the chain rule:


    \begin{gathered}<br />
  \frac{{dy}}<br />
{{dx}} = 6u\frac{{du}}<br />
{{dx}} + 2\frac{{du}}<br />
{{dx}} = \left( {6\sqrt {x^2  + 5}  + 2} \right)\frac{{du}}<br />
{{dx}} =  \hfill \\<br />
   \hfill \\<br />
   = \left( {6\sqrt {x^2  + 5}  + 2} \right)\left[ {x\left( {x^2  + 5} \right)^{ - 0.5} } \right] \hfill \\ <br />
\end{gathered} <br />
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  4. #4
    GAMMA Mathematics
    colby2152's Avatar
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    Quote Originally Posted by NAPA55 View Post
    3. The amount of pollution in a certain lake is P(t) = (t^0.25 + 3), where t is years and P is parts per million. What is the rate of change after 16 years (I know I use the limit formula but the exponents and t+h are confusing me).
    You don't need the limit "formula". Just find the derivative, and solve for when time is equal to 16.

    P'(t)=3(t^{0.25}+3)^2 (0.25t^{-0.75})

    P'(16)=3(25)+ (\frac{1}{32}) \Rightarrow  \frac{75}{32}

    Check my arithmetic, but the derivative is correct.
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