# How do you sketch the graph #y=sqrt(1+x^2)# using the first and second derivatives?

##### 1 Answer

#y=sqrt(1+x^2)=(1+x^2)^(1/2)#

Use the chain rule to differentiate:

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

Using the product and chain rules:

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

#color(white)((d^2y)/dx^2)=(1+x^2)^(-1/2)-x^2(1+x^2)^(-3/2)#

#color(white)((d^2y)/dx^2)=1/(1+x^2)^(1/2)-x^2/(1+x^2)^(3/2)#

#color(white)((d^2y)/dx^2)=(1+x^2-x^2)/(1+x^2)^(3/2)#

#color(white)((d^2y)/dx^2)=1/(1+x^2)^(3/2)#

The domain of

We see that

The function is decreasing on

We can also note that

The drawing of the graph should then look something like:

graph{sqrt(1+x^2) [-17.34, 18.7, -4.97, 13.05]}

The graph is always concave up, has a minimum at