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So i'm trying to remind myself of how to use the quotient rule, and when i try to differentiate the following problem i get major issues:

(2x + x^3) / x^(1/2)

I get to the following point...

((2 + 3x^3)(x^1/2) - 1/2(x^1/2)(2x+x^3)) / x

How in gods name do i simplify this?

the ultima
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2 Answers2

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The function can be rewritten in a simpler way: $$ f(x)=\frac{2x+x^3}{x^{1/2}}=2x^{1/2}+x^{5/2} $$ and the derivative is $$ f'(x)=2\cdot\frac{1}{2}x^{-1/2}+\frac{5}{2}x^{3/2}= \frac{1}{x^{1/2}}+\frac{5x^{3/2}}{2}= \frac{2+5x^2}{2x^{1/2}} $$

You may also use the quotient rule, if you really want it: $$ f'(x)=\frac{(2+3x^2)x^{1/2}-\dfrac{1}{2x^{1/2}}(2x+x^3)}{x}= \frac{4x+6x^3-2x-x^3}{2x\cdot x^{1/2}}= \frac{2x+5x^3}{2x\cdot x^{1/2}} $$ and, after simplifying, you get the same as above.

egreg
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It is easier using logarithmic differentiation:

Set $f(x)=\dfrac{2x+x^3}{x^{\frac12}}=\dfrac{x(2+x^2)}{x^{\frac12}}$. Then: $$\frac{f'(x)}{f(x)}=\frac1x+\frac{2x}{2+x^2}-\frac12\cdot\frac1x=\frac1{2x} +\frac{2x}{2+x^2}=\frac{2+5x^2}{2x(2+x^2)},$$ whence $$f'(x)=\frac{f'(x)}{f(x)}\cdot f(x)=\frac{2+5x^2}{2x^{\frac12}}.$$

Bernard
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