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Math Help - Finding the derivative

  1. #1
    Member Mr Rayon's Avatar
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    Finding the derivative

    Find the derivative of the following from first principles:

     <br />
x^3 + 2x<br />
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  2. #2
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    mr fantastic's Avatar
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    Quote Originally Posted by Mr Rayon View Post
    Find the derivative of the following from first principles:

     <br />
x^3 + 2x<br />
    Do you know the appropriate formula to be using? Where do you get stuck in applying it? Please show what work you've done and where you get stuck.

    (Aside: You will find many worked examples both at MHF and elsewhere).
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  3. #3
    Member Mr Rayon's Avatar
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    Quote Originally Posted by mr fantastic View Post
    Do you know the appropriate formula to be using? Where do you get stuck in applying it? Please show what work you've done and where you get stuck.
    Very well...

    f(x) = x^3 - 2x

    f(x + h) = (x + h)^3 + 2(x + h)

    f(x + h) - f(x) = x^3 + h^3 - 2x - 2h

    = x(x^2 - 2) + h(h^2 - 2)

    Substituting into, \lim_{h\to 0} \frac{f(x + h) - f(x)}{h}, we get:

    \lim_{h\to 0} \frac{x(x^2 - 2) + h(h^2 - 2)}{h}

    \lim_{h\to 0} x(x^2 - 2) + h^2 - 2, h\neq 0

    = x ^3 - 2x - 2


    Quote Originally Posted by mr fantastic View Post
    (Aside: You will find many worked examples both at MHF and elsewhere).
    Oh really? Could you send me the links?
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Mr Rayon View Post
    Very well...

    f(x) = x^3 - 2x

    f(x + h) = (x + h)^3 + 2(x + h)

    f(x + h) - f(x) = x^3 + h^3 - 2x - 2h

    = x(x^2 - 2) + h(h^2 - 2)

    Substituting into, \lim_{h\to 0} \frac{f(x + h) - f(x)}{h}, we get:

    \lim_{h\to 0} \frac{x(x^2 - 2) + h(h^2 - 2)}{h}

    \lim_{h\to 0} x(x^2 - 2) + h^2 - 2, h\neq 0

    = x ^3 - 2x - 2
    That's not correct.

    f(x+h)-f(x)\neq x(x^2 - 2) + h(h^2 - 2)!

    f(x+h)=(x+h)^3-2(x+h)=x^3+3x^2h+3xh^2+h^3-2x-2h

    Thus, f(x+h)-f(x)=x^3+3x^2h+3xh^2+h^3-2x-2h-x^3+2x=3x^2h+3xh^2+h^3-2h

    So what is \lim_{h\to0}\frac{f(x+h)-f(x)}{h} equal to now?
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