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.
Let's consider the function:
f(x) = (sin(x^2) + 2x) / (cos(3x) - x^2)
In order to differentiate this function, we will need to use the quotient rule, product rule, and chain rule. The order in which we will use these rules is as follows:
1) Chain rule: We will apply the chain rule first since we have a composite function within the numerator and denominator. Specifically, we will differentiate the inner functions sin(x^2) and cos(3x).
2) Product rule: We will apply the product rule next as we have two terms in the numerator to differentiate - sin(x^2) and 2x.
3) Quotient rule: Finally, we will use the quotient rule since we have a ratio of two differentiated functions.
To differentiate the function f(x) = (sin(x^2) + 2x) / (cos(3x) - x^2), we will follow this order of rules.