Answer
(a) True
(b) False
Work Step by Step
(a) The elementary functions are polynomials, rational functions, power functions, exponential functions, logarithmic functions, trigonometric functions, and the addition, subtraction, multiplication, division, and composition of such functions.
It follows from the chain rule that all such functions have derivatives that are elementary functions.
Hence, the given statement is true.
(b) The elementary functions are polynomials, rational functions, power functions, exponential functions, logarithmic functions, trigonometric functions, and the addition, subtraction, multiplication, division, and composition of such functions.
Since, $f(x)=e^{-x^{2}}$ is the composition of an exponential function and a polynomial function and therefore, an elementary function, yet it is well known to NOT have an anti-derivative.
Hence, the statement is false.
