How do you find the derivative of #x/ sqrt (x^2 +1)#?

1 Answer
Jan 26, 2017

Answer:

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

Explanation:

Using the quotient rule:

#d/(dx) (x/sqrt(x^2+1)) = (sqrt(x^2+1) *d/(dx) (x) - x *d/(dx) (sqrt(x^2+1)))/(sqrt(x^2+1))^2#

#d/(dx) (x/sqrt(x^2+1)) = (sqrt(x^2+1) - x *d/(dx) (sqrt(x^2+1)))/(x^2+1)#

We can now use the chain rule to calculate:

#d/(dx) (sqrt(x^2+1)) = 2x* 1/(2sqrt(x^2+1)) = x/sqrt(x^2+1)#

and substitute it in the expression above:

#d/(dx) (x/sqrt(x^2+1)) = (sqrt(x^2+1) - x *x/sqrt(x^2+1))/(x^2+1)#

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

#d/(dx) (x/sqrt(x^2+1)) = 1/((x^2+1)sqrt(x^2+1))#