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Thread: 2nd Deriv to Pt Inflect (number 2)

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    2nd Deriv to Pt Inflect (number 2)

    3x/(x+8).

    I need to use the 2nd derivative test to find the point(s) of inflection. I get confused about how to do derivatives of examples like this.
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    Quote Originally Posted by akuczma86 View Post
    3x/(x+8).

    I need to use the 2nd derivative test to find the point(s) of inflection. I get confused about how to do derivatives of examples like this.
    Quotient rule

    If $\displaystyle f(x) = \frac{x}{y}$

    then, $\displaystyle f'(x) = \frac{x'y-y'x}{y^2}$

    This should definitely be in your books/notes
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    So the first derivative would be -3x/(x+8)^2?
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    MHF Contributor harish21's Avatar
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    Quote Originally Posted by akuczma86 View Post
    So the first derivative would be -3x/(x+8)^2?
    NO.

    Its

    $\displaystyle \frac {([\text{derivative of 3x}]\times (x+8)) - (3x \times [\text{derivative of} (x+8)])}{(x+8)^2}$
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    Quote Originally Posted by akuczma86 View Post
    So the first derivative would be -3x/(x+8)^2? <<<<<< no
    $\displaystyle f(x)=\frac{3x}{x+8}$

    $\displaystyle f'(x)=\frac{(x+8) \cdot 3 -3x \cdot 1}{(x+8)^2}$

    Expand the bracket in the numerator, collect like terms.
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    $\displaystyle f'(x)= 24/ (x^2 + 16x + 64$

    $\displaystyle f''(x)= (-24 -3x)/(x^4 + 2x^3 + 19x^2 + 48x + 36)$

    Am I right here?
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    It is OK for f'(x)

    BTW an easiest way in this case is $\displaystyle f(x) = \frac{3x}{x+8} = 3 - \frac{24}{x+8}$. The derivative of f is the same as the derivative of $\displaystyle -\frac{24}{x+8}$ which is $\displaystyle \frac{24}{(x+8)^2}$

    The derivative of $\displaystyle u^n(x)$ is $\displaystyle n u'(x) u^{n-1}(x)$

    Use this formula with u(x) = x+8 and n=-2 to get the derivative of $\displaystyle \frac{1}{(x+8)^2}$ then multiply by 24
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    Quote Originally Posted by akuczma86 View Post
    $\displaystyle f'(x)= 24/ (x^2 + 16x + 64$

    $\displaystyle f''(x)= (-24 -3x)/(x^4 + 2x^3 + 19x^2 + 48x + 36)$

    Am I right here? <<<<<< I can't imagine how you got this result
    $\displaystyle f'(x)= \frac{24}{x^2 + 16x + 64}$ then

    $\displaystyle f''(x)= \frac{(x^2+16x+64) \cdot 0 - 24(2x+16)}{ (x^2 + 16x + 64)^2} = \boxed{\frac{ - 48x-384}{ x^4 + 32x^3 + 384x^2 + 2048x + 4096} }$

    This result is correct but nearly useless. Compare with the method posted by running-gag:

    $\displaystyle f'(x)= \frac{24}{x^2 + 16x + 64} = 24 (x+8)^{-2}$ then

    $\displaystyle f'(x)= -48 (x+8)^{-3} \cdot 1 =\boxed{ -\frac{48}{(x+8)^3}}$

    Both terms describe the same function!
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