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. find the derivative of it
Let's say we have the function:
\[ f(x) = \sin(x) \cdot e^x \cdot \sqrt{3x^2 +6} \]
To find the derivative of this function, we need to apply the quotient rule, product rule, and chain rule. The order in which we apply these rules is as follows:
1. Chain Rule
2. Product Rule
3. Quotient Rule
Now, let's differentiate the function:
1. Chain Rule:
First, we differentiate the outermost function using the chain rule. The derivative of \(\sin(x)\) is \(\cos(x)\). Everything else remains the same:
\[ f'(x) = \cos(x) \cdot e^x \cdot \sqrt{3x^2 + 6} \]
2. Product Rule:
Next, we apply the product rule to differentiate the remaining two factors: \(\cos(x)\cdot e^x\) and \(\sqrt{3x^2 + 6}\). The product rule states that the derivative of a product of functions is the derivative of the first function times the second function, plus the first function times the derivative of the second function. Applying the product rule to our function:
\[ f'(x) = \cos(x) \cdot e^x \cdot \sqrt{3x^2 + 6} + \sin(x) \cdot e^x \cdot \frac{d}{dx}(\sqrt{3x^2 + 6}) \]
3. Quotient Rule:
Finally, we apply the quotient rule to differentiate \(\frac{d}{dx}(\sqrt{3x^2 + 6})\). The quotient rule states that the derivative of a quotient of functions is the derivative of the numerator times the denominator, minus the numerator times the derivative of the denominator, divided by the square of the denominator. Applying the quotient rule:
\[ f'(x) = \cos(x) \cdot e^x \cdot \sqrt{3x^2 + 6} + \sin(x) \cdot e^x \cdot \left(\frac{(6x)}{2\sqrt{3x^2 + 6}}\right) \]
Note: The above expression gives us the derivative of \(f(x)\) by applying the quotient rule, product rule, and chain rule in the order mentioned. However, we did not actually calculate the final derivative value, as it was not required in the question.