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 dont need to differentiate your function.
Let's consider the function f(x) = (sin(x) + x^2) / (3x).
To differentiate this function, we need to use the quotient rule, product rule, and chain rule. The order in which we apply the three rules is as follows:
1. Chain Rule: We will start by applying the chain rule to differentiate the sin(x) term within the numerator. This means we differentiate the sin(x) function and multiply it by the derivative of the inner function x, which is just 1.
- Differentiation of sin(x): cos(x)
2. Product Rule: Next, we will apply the product rule to differentiate the numerator, which is the sum of sin(x) and x^2. The product rule states that the derivative of the product of two functions is equal to the derivative of the first function times the second function plus the first function times the derivative of the second function.
- Differentiation of sin(x) term: cos(x) * 1 = cos(x)
- Differentiation of x^2 term: 2x
3. Quotient Rule: Finally, we will apply the quotient rule to differentiate the entire function. The quotient rule states that the derivative of the division of two functions is equal to the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.
- Differentiation of the numerator (sin(x) + x^2): cos(x) + 2x
- Differentiation of the denominator (3x): 3
- Differentiated function: [ (cos(x) + 2x) * (3x) - (sin(x) + x^2) * 3] / (3x)^2
Remember, this example explains the process of applying the quotient rule, product rule, and chain rule to differentiate the given function. The resulting function after differentiation is not provided here.