Product Rule
When it comes to differentiating functions in calculus, one of our first tools is the product rule. This rule is crucial when you're working with functions that are products of two or more other functions. It states that to find the derivative of a product of two functions, we differentiate one function and multiply it by the other function as is, then add the product of the first function and the derivative of the second function.
For example, if you have a function like \( u(x)v(x) \), where both \( u \) and \( v \) are differentiable functions of \( x \), then the derivative of the product, \( u(x)v(x) \), would be \( u'(x)v(x) + u(x)v'(x) \). In our exercise, \( f(x) = 2 x \sqrt{9-x^{2}} \), we can think of \( 2x \) as one function (let’s call it \( u \)) and \( \sqrt{9-x^{2}} \) as the second function (let’s call it \( v \)). Using the product rule, we differentiate each and add them together to get our derivative.
Chain Rule
Another essential rule in calculus for finding derivatives is the chain rule. This rule is used when you have functions composed of other functions, often referred to as composite functions. The chain rule allows us to take the derivative of the outer function and then multiply it by the derivative of the inner function.
The generic form of the chain rule is expressed as if you have a composite function \( h(x) = f(g(x)) \), then the derivative \( h'(x) = f'(g(x))*g'(x) \). In the context of our problem, when differentiating \( \sqrt{9-x^{2}} \), we treat \( 9-x^{2} \) as the inner function, \( g(x) \), and \( \sqrt{\cdot} \) as the outer function, \( f \). Using the chain rule, we differentiate \( \sqrt{\cdot} \) which gives us \( \frac{1}{2\sqrt{\cdot}} \), and then multiply by the derivative of the inner function, which is \( -2x \).
Critical Numbers
Critical numbers play a pivotal role in analyzing functions. They are the \( x \)-values where the function's derivative is zero or undefined, which usually correspond to local minima, local maxima, or inflection points in the function's graph. Finding critical numbers is a step towards understanding the shape and behavior of a function.
For our function \( f(x) \), we find the derivative and set it equal to zero to find the critical numbers, in this case \( x=0 \). Since the derivative is also undefined at \( x=\pm3 \), these could be critical points as well, but we're only considering the open interval \( (-3,3) \), so \( x=\pm3 \) are not included. The information about critical numbers helps us to further analyze the function, particularly for graphing and understanding the intervals of increase and decrease.
Function Graphing
Understanding how to graph functions and their derivatives can provide great insights into their behavior. When graphing functions, we look for key features: where the function increases and decreases, local maxima and minima, inflection points, and asymptotes. The derivative of the function can help us determine many of these features.
In the exercise, after computing the derivative \( f'(x) \), we use it to determine the intervals where the function is increasing and decreasing. Graphs of \( f(x) \) and \( f'(x) \) are often sketched on the same axes to provide a visual comparison: where \( f'(x) > 0 \), the function increases, and where \( f'(x) < 0 \), it decreases. In our case, the graph of \( f \) resembles a hill. By comparing this to the sign of \( f'(x) \), we can predict the behavior and shape of \( f \) without even looking at the graph.
