How do you find the second derivative of #z=(x-5)/(x+2)#?

2 Answers
Andrea S.
Mar 31, 2017

#(d^2z)/dx^2 = -14/(x+2)^3 #

Explanation:

Calculate the first derivative using the quotient rule:

#(dz)/dx = ((x+2)d/dx(x-5) -(x-5)d/dx(x+2))/(x+2)^2#

#(dz)/dx = (x+2 -x+5)/(x+2)^2 = 7/(x+2)^2#

Then using the chain rule:

#(d^2z)/dx^2 = d/dx (7/(x+2)^2) = -14/(x+2)^3 d/dx (x+2) = -14/(x+2)^3 #

Alternatively you can write the function as:

#z= (x-5)/(x+2) = (x+2-7)/(x+2) = 1-7/(x+2) = 1-7(x+2)^-1#

and you can see that:

#(d^nz)/dx^n = 7(-1)^(n+1)(n!)/(x+2)^(n+1)#

Jim G.
Mar 31, 2017

#(d^2z)/dx^2=(-14)/(x+2)^3#

Explanation:

One way is to express the function as a product and differentiate using the product rule.

#rArrz=(x-5)/(x+2)=(x-5)(x+2)^-1#

differentiate using the #color(blue)"product rule"#

#"Given "z=g(x)h(x)" then"#

#color(red)(bar(ul(|color(white)(2/2)color(black)(dz/dx=g(x)h'(x)+h(x)g'(x))color(white)(2/2)|)))#

#rArrdz/dx=(x-5).-1(x+2)^-2+(x+2)^(-1).1#

#"to obtain "(d^2z)/(dx^2)" differentiate " dz/dx#

differentiate the first term using the #color(blue)"product rule"#

#rArr(d^2z)/(dx^2)=(x-5).2(x+2)^-3-(x+2)^(-2).1-(x+2)^-2#

#=2(x-5)(x+2)^-3-2(x+2)^-2#

#=2(x+2)^-3(x-5-x-2)#

#=-14/(x+2)^3#

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