Find the second derivative?? #(2x+5)/(3x-2)#

1 Answer
Jim G.
Nov 8, 2017

#114/(3x-2)^3#

Explanation:

#"let "f(x)=(2x+5)/(3x-2)#

#"to find the first derivative differentiate using the "color(blue)"quotient rule"#

#"given "f(x)=(g(x))/(h(x))" then"#

#f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x))^2larr"quotient rule"#

#g(x)=2x+5rArrg'(x)=2#

#h(x)=3x-2rArrh'(x)=3#

#rArrf'(x)=((3x-2)2-(2x+5)3)/(3x-2)^2#

#color(white)(rArrf'(x))=(6x-4-6x-15)/(3x-2)^2=(-19)/(3x-2)^2#

#"to obtain the second derivative , differentiate the first"#
#"derivative"#

#"either use the quotient rule again or use the "color(blue)"chain rule"#

#"given "f(x)=g(h(x))" then"#

#f'(x)=g'(h(x))xxh'(x)larr"chain rule"#

#f'(x)=(-19)/(3x-2)^2=-19(3x-2)^(-2)#

#rArrf''(x)=-19xx-2(3x-2)^(-3).3#

#color(white)(rArrf''(x))=114/(3x-2)^3#

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