Let f(x)=e^rx and g(x)=e^sx , for constants r and s.

(a) Calculate f'(x)g'(x) and (f(x)g(x))'.

(b) How must r be related to s (i.e., find the equation that they must obey) so that your two answers to Problem (a) are equal? In other words, when can the product of the derivatives equal the derivative of the product? Identify two specific pairs of numerical values for r and s that satisfy your question.

(c) Calculate f'(x)/g'(x) and (f(x)/g(x))'.

(d) How must r be related to s (i.e., find the equation that they must obey) so that your two answers to Problem (c) are equal? In other words, when can the quotient of the derivatives equal the derivative of the quotient? Identify two specific pairs of numerical values for r and s that satisfy your equation.

(e) Is it possible that simultaneously, for a single pair of r and s values, the product of the derivatives equals the derivative of the product and the quotient of the derivatives equals the derivative of the quotient. Explain.