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

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Answer 1 · 2017-03-09T20:23:32

#(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)))#