How to find the second derivative of $f(x)=\frac{x^2}{10+x}$

Question

How do you find the second derivative of $f(x)=\frac{x^2}{10+x}$?

I get $f'(x)={20x+x^2}/{(10+x)^2}$

Then get stuck here:

$(10+x)^2 (20+2x) - (20x+x^2) \frac {2(10+x)}{((10+x)^2)^2}$

Answer 1

First make the Euclidean division: $$x^2=(x-10)(x+10)+100, \enspace\text{whence}\quad\frac{x^2}{x+10}=x-10+\frac{100}{x+10}$$ Thus \begin{align} f'(x)&=1-\frac{100}{(x+10)^2},\\ f''(x)&=\frac{200}{(x+10)^3}. \end{align}

Answer 2

f(x) = $\frac{x^2}{10+x}$

You need to use the quotient rule: $(\frac{f}{g})'=\frac{f'*g-g'*f}{g^2}$

Then f= $x^2$ take the derivative of that and you get $\rightarrow f'=2x$

$g= 10+x$, take the derivative of that and you get $\rightarrow g'= 1$

Now let us apply the quotient rule and use our following notations.

$f'(x)= \frac{(2x)*10+x - ((1)*x^2)}{(10+x)^2}$

Now let us simplify and we get $\rightarrow \frac{20x+x^2}{(10+x)^2}$

Let us get $f''(x)$, but first let us get the notations ready to substitute them.

$f'= 20x+x^2 \rightarrow f''=20+2x$ & $g'= (10+x)^2 \rightarrow g''= 2(10+x)$

Then we use the quotient rule again and we get:

$f''(x)= \frac{(2x+10)*(10+x)^2 - (20x+x^2(2(10+x))}{((10+x)^2)^2}$

We simplify and get $\rightarrow f''(x)= \frac{200}{(x+10)^3}$

Answer 3

Just do it:

$f(x) = \frac {x^2}{10 - x}$

$f'(x) = \frac {2x(10 - x) - x^2(-1)}{(10-x)^2} = \frac{-2x^2 + 20x +x^2}{(10-x)^2} = \frac{20x - x^2}{(10 - x)^2}=\frac {x(20-x)}{(10-x)^2}$

$f''(x) =\frac{[x(-1)+ (20-x)](10-x)^2 - x(20 -x)2(10-x)(-1)}{(10-x)^4}$

$=\frac{[x(-1)+ (20-x)](10-x)^{\not 2} - x(20 -x)2\not{(10-x)}(-1)}{(10-x)^{\not 43}}$

$=\frac{[-x+ (20-x)](10-x) + 2x(20 -x)}{(10-x)^3}$

$=\frac{2(10-x)^2 + 2x(20-x)}{(10-x)^3}=\frac{200 - 40x + 2x^2 + 40x - 2x^2}{(10-x)^3}=\frac{200}{(10-x)^3}$

Or you can generalize first:

$(\frac fg)'' =$

$((\frac fg)')' = $

$(\frac{f'g - fg'}{g^2})'=$

$(\frac{f'}g)' - (\frac{fg'}{g^2}) =$

$\frac{f''g - f'g'}{g^2} - \frac{(f'g'+fg'')g^2 - fg'*2gg'}{g^4} =$

$\frac{f''}g - \frac{f'g'}{g^2} - \frac{f'g'}{g^2} - \frac{fg''}{g^2} + \frac{2fg'^2}{g^3}=$

$\frac{f''}g - 2\frac{f'g'}{g^2} - \frac{fg''}{g^2} + \frac{2fg'^2}{g^3}=$

$\frac{2}{10-x} - 2\frac{-2x}{(10-x)^2} - 0 + \frac{2x^2}{(10-x)^3}=$

$\frac {2(10-x)^2 + 2x(10 - x) + 2x^2}{(10-x)^3}=$

$\frac{2[(10-x)+x]^2}{(10-x)^3 }=\frac {2*10^2}{(10-x)^3}= \frac {200}{(10-x)^2}$

But I can't see that $(\frac fg)'' = \frac{f''}g - 2\frac{f'g'}{g^2} - \frac{fg''}{g^2} + \frac{2fg'^2}{g^3}$ is anything worth memorizing.... but still....