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Math Help - A few Questions on the derivative

  1. #1
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    A few Questions on the derivative

    Hello, I had a few questions that I completed and I'm unsure if my answers are correct. i had to find the derivative of the √2x, using first principle, f(a+h) - f(a)/h. The answer that I arrived at is 1/√2√x, but when i use the power rule i get 1/x^2. Are they both wrong, or am i doing something wrong, somewhere? The second question is finding the derivative of 3x/(x^2 + 4), the answer i arrived at, is 3/x^2 + 4 - 6x^2/(x^2 + 4)^2. I got this using the quotient rule. Sorry about the mess, i'm completly new to calculus, have no one around me to check my answers until monday and, I just want to know my mistakes. Again, thanks very much.
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  2. #2
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    Hi.

    Quote Originally Posted by walk_in_fark View Post
    Hello, I had a few questions that I completed and I'm unsure if my answers are correct. i had to find the derivative of the √2x, using first principle, f(a+h) - f(a)/h. The answer that I arrived at is 1/√2√x, but when i use the power rule i get 1/x^2. Are they both wrong, or am i doing something wrong, somewhere?
    No, the first solution is correct, I guess you mean f'(x) = \frac{1}{\sqrt 2 \sqrt x}, don't you?

    f(x) = \sqrt{2x} = \sqrt{2} \sqrt{x} = \sqrt{2} x^{1/2}

    Thus
    f'(x) = \sqrt 2 \frac 1 2 x^{-1/2} =\frac{ \sqrt 2}{2} * \frac{1}{x^{0.5}}

    =\frac{1}{\sqrt 2 \sqrt{x}}




    Quote Originally Posted by walk_in_fark View Post
    The second question is finding the derivative of 3x/(x^2 + 4), the answer i arrived at, is 3/x^2 + 4 - 6x^2/(x^2 + 4)^2. I got this using the quotient rule. Sorry about the mess, i'm completly new to calculus, have no one around me to check my answers until monday and, I just want to know my mistakes. Again, thanks very much.
    Yes, that is correct, but I prefer

    f(x) = \frac{3x}{x^2+4}

    f'(x) = \frac{3(4 - x^2)}{(x^2 + 4)^2} (this is the same solution...)

    kind regards
    Rapha
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  3. #3
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by walk_in_fark View Post
    Hello, I had a few questions that I completed and I'm unsure if my answers are correct. i had to find the derivative of the √2x, using first principle, f(a+h) - f(a)/h. The answer that I arrived at is 1/√2√x, but when i use the power rule i get 1/x^2. Are they both wrong, or am i doing something wrong, somewhere? The second question is finding the derivative of 3x/(x^2 + 4), the answer i arrived at, is 3/x^2 + 4 - 6x^2/(x^2 + 4)^2. I got this using the quotient rule. Sorry about the mess, i'm completly new to calculus, have no one around me to check my answers until monday and, I just want to know my mistakes. Again, thanks very much.
    first one
    I prefer to write it like

    \sqrt{2x}=(2x)^{\frac{1}{2}}

    \lim_{h\rightarrow0}\frac{f(a+h)-f(a)}{h}

    \lim_{h\rightarrow0}\frac{\sqrt{x+h}-\sqrt{x}}{h}=\lim_{h\rightarrow0}\left(\frac{\sqrt  {x+h}-\sqrt{x}}{h}\right)\left(\frac{\sqrt{x+h}+\sqrt{x}  }{\sqrt{x+h}+\sqrt{x}}\right)

    \lim_{h\rightarrow0}\left(\frac{x+h-x}{h(\sqrt{x+h}+\sqrt{x})}\right)=\lim_{h\rightarr  ow0}\left(\frac{h}{h(\sqrt{x+h}+\sqrt{x})}\right)

    the rest for you just sub zero instead of h after delete h from the denominator and the numentor
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  4. #4
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by walk_in_fark View Post
    3x/(x^2 + 4), the answer i arrived at, is 3/x^2 + 4 - 6x^2/(x^2 + 4)^2. .
    f(x)=\frac{3x}{x^2+4}=(3x)(x^2+4)^{-1}

    use product rule

    H(x)=g(x)f(x)....H'(x)=g'(x)f(x)+f'(x)g(x)
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