Final answer:
To find the derivative of a quotient of two functions, we can use the quotient rule.
This rule states that the derivative of a quotient of two functions is equal to the numerator times the derivative of the denominator minus the denominator times the derivative of the numerator, all divided by the square of the denominator.
Step-by-step explanation:
A) Quotient of two polynomials:
To find the derivative of the quotient of two polynomials, we can use the quotient rule. The quotient rule states that if we have a function f(x) = g(x)/h(x), then its derivative is given by f'(x) = [g'(x) * h(x) - g(x) * h'(x)] / [h(x)]^2.
B) Quotient of two logarithmic functions:
To find the derivative of the quotient of two logarithmic functions, we can use the quotient rule as well. The quotient rule remains the same as explained in the previous case. We just need to differentiate the logarithmic functions according to the rules of logarithmic differentiation.
C) Quotient of two exponential functions:
To find the derivative of the quotient of two exponential functions, we can once again use the quotient rule. The derivative of an exponential function is given by the product of the function itself and the natural logarithm of the base. Apply the quotient rule to differentiate the quotient of two exponential functions.
D) Quotient of two trigonometric functions:
To find the derivative of the quotient of two trigonometric functions, we can apply the quotient rule. The quotient rule for trigonometric functions is similar to the quotient rule used in previous cases. Differentiate the numerator and denominator using the rules for trigonometric differentiation, and then apply the quotient rule to find the final derivative.