How do you differentiate #f(x)=e^(sqrt(x^2+2))#?

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Answer 1 · 2018-03-15T13:02:45

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

We use the chain rule, which states that

#(df)/dx=(df)/(du)*(du)/dx#

Let #u=sqrt(x^2+2)#. Now we find differentiate #sqrt(x^2+2)# to find #(du)/dx#.

We use the chain rule again, and so #t=x^2+2, (dt)/dx=2x#, #f=sqrt(t), (df)/(dt)=1/(2sqrt(t)#.

Then, #(df)/dx=2x*1/(2sqrt(t)#

#=x/(sqrt(t)#

Reversing the substitution that #t=x^2+2#, we get

#=x/sqrt(x^2+2)# or #(du)/dx=x/sqrt(x^2+2)#

Now, we got #f=e^u, (df)/(du)=e^u#. Combining together, we get

#(df)/dx=e^u*x/sqrt(x^2+2)#

#=(xe^u)/sqrt(x^2+2)#

Reversing the substitution that #u=sqrt(x^2+2)#, we have

#=(xe^(sqrt(x^2+2)))/(sqrt(x^2+2))#

That's the answer.