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Determine the derivative.$$f(x)=\frac{e^{2 x^{2}}+1}{x^{2}+1}$$

$\frac{2 x\left[\left(2 x^{2}+1\right) e^{2 x^{2}}-1\right]}{\left(x^{2}+1\right)^{2}}$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 4

The Derivative of the Exponential Function

Campbell University

Oregon State University

Idaho State University

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$$\text { } , \text { …

we're finding the derivative of this other quotient. Eso. The big thing is that we have a quotient rule, but also inside the question. Or we're gonna have the chain role because we have e to the two x squared. That's gonna be a change whole plus one and in the denominator way of X squared plus one, uh, which is not really anything special. It's just part of the problem. So when you take the derivative using the quotient movement, you do the derivative of the top, which the derivative of E to the two X squared is E to the two X squared times, the derivative of two X squares, which would be four X and the derivative of 10 So we don't have to worry about that on any of that X squared plus one, you leave the denominator alone, uh, minus. Now you leave the numerator alone, uh, times the derivative of the denominator, which is two X all over the denominator squared, and some teachers will let you leave. Your answer like that on day is really not a benefit of simplifying it all. But if you wanted to, the numerator has the shared factor of two x eso If you wanted to, you could factor out a two x on. Then you'd be left with the quantity of two because two times to give you four Either the two x squared leaving that X squared plus one alone minus the quantity. Uh, or you could distribute that minus in there And, uh, all over that denominator squared eso This is another acceptable answer. So it's up to you one over the other or both. I don't care.

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