# Identify each of the following functions as polynomial, rational, algebraic, or transcendental f(x)=(2x^3+3x)/(9-7x^2)

Question
Rational functions
Identify each of the following functions as polynomial, rational, algebraic, or transcendental
$$f(x)=(2x^3+3x)/(9-7x^2)$$

2020-11-06
If a function can be defined by a rational fraction i.e. an algebraic fraction such that both the numerator and the denominator are polynomials then it is called ration function.
The given function has polynomial in numerator and denominator both, so this function is a rational function

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My answer. The intervals do match and equal zero so Rolles theorem can work.
Second I found the derivative maybe thats where I can't solve this problem.
The derivative that I got was $$\displaystyle{64}-{x}^{{2}}+\frac{{x}}{\sqrt{{{64}-{x}^{{2}}}}}$$ maybe i did wrong on the simplifying. I at least tried hopefully some one can explain as much as possible with every single step because I can figure out the algebra part.
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