give an example of a function where you must use the quotient rule, product rule, and chain rule one time each to differentiate it. State the order of which rule is used first, second, and third. Your example must contain a trigonometric function within it. you do not need to differentiate your function.
Consider the following function:
f(x) = (sin(x) + x^2) * e^x / (2x^2 + cos(x))
To differentiate this function, we need to apply the quotient rule, product rule, and chain rule. The order of the rules used can be:
1. Chain rule: We start by applying the chain rule since we have the function e^x within the expression.
2. Product rule: After taking the derivative of the exponential function, we can apply the product rule to differentiate the remaining terms (sin(x) + x^2) * e^x.
3. Quotient rule: Finally, we can apply the quotient rule to differentiate the entire function by dividing the derivative from the product rule by the derivative of the denominator (2x^2 + cos(x)).
The specific order of rules used can vary depending on the function, but in this example, we would use the chain rule first, followed by the product rule, and finally the quotient rule.