Question #1590c

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Answer 1 · 2018-01-09T14:19:42

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

Your original function is

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

For simplicity, I will rewrite this as the following

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

So this uses a chain rule and a power rule. The chain rule says the following:

#d/dx(f(g(x)))=f'(g(x))*g'(x)#

In this case, we can make the observation that

#f(x)=x^(1/2)#

and

#g(x) = 1+x^2#

So now we need to take the derivative of each:

#f'(x)=d/dx(x^(1/2))#

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

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

#g'(x)=2x#

So now we can finish the chain rule using the logic mentioned above that

#d/dx(f(g(x)))=f'(g(x))*g'(x)#

to get the final answer

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