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

##### 2 Answers

#### Explanation:

Calculate the first derivative using the quotient rule:

Then using the chain rule:

Alternatively you can write the function as:

and you can see that:

#### 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#