# taking the second derivative of a rational function

#### kingsolomonsgrave

Taking the second derivative of $$\displaystyle x/(x+1)^{4}$$

for the first derivative I got

$$\displaystyle 1-3x/(x+1)^5$$

the second derivative I'm not sure about. Im not even sure I did the first one correctly.

thanks!

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#### TheEmptySet

MHF Hall of Honor
Taking the second derivative of $$\displaystyle x/(x+1)^{4}$$

for the first derivative I got

$$\displaystyle ((x+1)^3-4x)/(p+1)^7$$

the second derivative I'm not sure about. Im not even sure I did the first one correctly.

thanks!

No the first derivative is not correct.

I am not a fan of the quotient rule, so I don't usually use it. Just use negative exponents to rewrite and use the product rule.

$$\displaystyle x(x+1)^{-4}=(x+1)^{-4}-4x(x+1)^{-5}=(x+1)^{-5}((x+1)-4x)=\frac{-3x+1}{(x+1)^5}$$

#### Plato

MHF Helper
Taking the second derivative of $$\displaystyle x/(x+1)^{4}$$
Write it as $$\displaystyle y=x(x+1)^{-4}$$
Then $$\displaystyle y'=(x+1)^{-4}+x[-4(x+1)^{-5}]$$

What is next?

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#### wasp

Taking the second derivative of $$\displaystyle x/(x+1)^{4}$$

for the first derivative I got

$$\displaystyle 1-3x/(x+1)^5$$

the second derivative I'm not sure about. Im not even sure I did the first one correctly.

thanks!
Your first derivative is correct, though you should write it properly as
$$\displaystyle f'(x) = \frac{-3x + 1}{(x + 1)^5}$$

So now do your quotient rule again:
$$\displaystyle f''(x) = \frac{(-3)(x + 1)^5 - (-3x + 1)*5(x + 1)^4}{(x + 1)^{10}}$$

-Dan

FYI By order of operations
$$\displaystyle 1 -3x / (x + 1)^5 = 1 - \frac{3x}{(x + 1)^5}$$

#### Salahuddin559

Hi Plato, there is a small (probably typing error) in your answer for f'.

Salahuddin
Maths online