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.

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