What is the second derivative of $f (x) = \frac{\sqrt{5 + x^{6}}}{x}$ $f(x)= sqrt(5+x^6)/x$ ?

Question

2347 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-03-09T20:23:32

$\frac{2 x^{12} + 65 x^{6} + 50}{x^{3} (5 + x^{6}) (\sqrt{5 + x^{6}})}$ $(2x^12+65x^6+50)/(x^3(5+x^6)(sqrt(5+x^6)))$

The second derivative is the derivative of the first derivative:

$f'(x)=(1/(cancel2sqrt(5+x^6))*cancel6^3x^5*x-sqrt(5+x^6)*1)/x^2$

$=((3x^6)/(sqrt(5+x^6))-sqrt(5+x^6))/x^2$

$=((3x^6-5-x^6)/(sqrt(5+x^6)))/x^2$

$=(2x^6-5)/(x^2sqrt(5+x^6))$

$f''(x)=(12x^5*x^2sqrt(5+x^6)-(2x^6-5)*(2xsqrt(5+x^6)+x^2/(cancel2sqrt(5+x^6))*cancel6^3x^5))/(x^4(5+x^6)$

$=(12x^7sqrt(5+x^6)-(2x^6-5)*(2xsqrt(5+x^6)+(3x^7)/(sqrt(5+x^6))))/(x^4(5+x^6)$

$=(12x^7sqrt(5+x^6)-(2x^6-5)*(2x(5+x^6)+3x^7)/(sqrt(5+x^6)))/(x^4(5+x^6))$

$=(12x^7sqrt(5+x^6)-(2x^6-5)*(10x+2x^7+3x^7)/(sqrt(5+x^6)))/(x^4(5+x^6))$

$=(12x^7sqrt(5+x^6)-(2x^6-5)*(10x+5x^7)/(sqrt(5+x^6)))/(x^4(5+x^6))$

$=((12x^7(5+x^6)-(2x^6-5)*(10x+5x^7))/(sqrt(5+x^6)))/(x^4(5+x^6))$

$=(12x^7(5+x^6)-(2x^6-5)*(10x+5x^7))/(x^4(5+x^6)(sqrt(5+x^6)))$

$=(60x^7+12x^13-20x^7-10x^13+50x+25x^7)/(x^4(5+x^6)(sqrt(5+x^6)))$

$=(2x^13+65x^7+50x)/(x^4(5+x^6)(sqrt(5+x^6)))$

$=(2x^12+65x^6+50)/(x^3(5+x^6)(sqrt(5+x^6)))$

What is the second derivative of f(x)= sqrt(5+x^6)/xf(x)=√5+x6xf(x)= sqrt(5+x^6)/x?