Question #0abe4

1 Answer
Jul 30, 2017

#d/(dx)[(sqrt(x+1))/x^2]=1/(2x^2sqrt(x+1))-(2sqrt(x+1))/x^3#

Explanation:

The Quotient Rule States

#[(f(x))/(g(x))]'=(f'(x)g(x)-f(x)g'(x))/(g(x)^2)#

In this case

#f(x)=sqrt(x+1)#

and

#g(x)=x^2#

To find #f'(x)# we will use the Chain Rule, which states

#[h(k(x))]'=h'(k(x))k'(x)#

In this case

#h(x)=sqrt(x)#

and

#k(x)=x+1#

Since the derivative of a constant is #0# and the derivative of #x# is #1#

#k'(x)=1#

Also remember that #sqrt(x)=x^(1/2)#

and the Power rule tells us that #(x^n)'=nx^(n-1)#

Then

#h'(x)=1/2x^(1/2-1)=1/2x^(-1/2)=1/(2sqrt(x)#

Then we plug in and find that

#f'(x)=1/(2sqrt(x+1))#

and we can use the power rule again to find #g'(x)#

#g'(x)=(x^2)'=2x^(2-1)=2x^1=2x#

Then we plug in

#(sqrt(x+1)/(x^2))'=((x^2)/(2sqrt(x+1))-2xsqrt(x+1))/(x^2)^2#

and we simplify

#=((x^2)/(2sqrt(x+1))-2xsqrt(x+1))/(x^4)#

#=((x^2)/(2sqrt(x+1)))x^-4-(2xsqrt(x+1))/x^4#

#=x^-2/(2sqrt(x+1))-(2sqrt(x+1))/x^3#

#=1/(2x^2sqrt(x+1))-(2sqrt(x+1))/x^3#