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An algebraic function is any function that can be constructed using a finite number of algebraic operations, which include addition, subtraction, multiplication, division, and taking roots, on variables and constants. This means algebraic functions can be as simple as a linear function like \(f(x) = 2x + 1\) or more complex expressions like \(f(x) = x\sqrt{x^2 + 1}\). In our original exercise, the function \(f(x) = x\left(1-\frac{4}{x+3}\right)\) is an example of an algebraic function. It includes basic operations of multiplication, subtraction, and division. While looking to differnetiate such a function, it is often helpful first to simplify it. This is exactly what was done in Step 1 by rewriting our function as a single expression. By simplifying the function, we make the task of finding a derivative more straightforward, reducing the chances of mistakes when using differentiation rules.

The product rule is a fundamental principle in calculus for finding the derivative of a product of two functions. If you have a function defined as the product of two functions, say \(f(x) = g(x) \, h(x)\), the product rule states that the derivative \(f'(x)\) is: \[ f'(x) = g'(x) \, h(x) + g(x) \, h'(x) \] This means that you take the derivative of the first function, multiply it by the second function, and then add the product of the first function and the derivative of the second function. In our original exercise, the product rule was employed when deriving the second term: \(x\left(-\frac{4}{x+3}\right)\). Here, the first function is \(u = x\) and the second function is \(v = -\frac{4}{x+3}\). By applying the product rule:

• The derivative of \(x\) is \(1\).
• The derivative of \(-\frac{4}{x+3}\) needs the quotient rule (as it is a fraction), and is thus calculated separately beforehand.
• Combine according to the product rule \(u'v + uv'\).

The quotient rule is an essential tool in calculus used to compute the derivative of a quotient of two functions. If a function \(f(x)\) is defined as the quotient \(f(x) = \frac{g(x)}{h(x)}\), the quotient rule provides that the derivative \(f'(x)\) is given by: \[ f'(x) = \frac{g'(x) \, h(x) - g(x) \, h'(x)}{(h(x))^2} \] In simpler terms, this formula says to:

• Take the derivative of the numerator \(g(x)\) and multiply it by the denominator \(h(x)\).
• Subtract from this the product of the numerator \(g(x)\) and the derivative of the denominator \(h(x)\).
• Divide the whole expression by the square of the denominator \((h(x))^2\).

In our exercise, the quotient rule was utilized to find the derivative of \(\frac{4}{x+3}\). Let \(g(x) = 4\) and \(h(x) = x+3\):

• \(g'(x)\) becomes \(0\) because it is a constant.
• \(h'(x)\) is \(1\).
• Applying the rule yields \(-\frac{4}{(x+3)^2}\).

By mastering the product and quotient rules, you can easily tackle complex functions like the one in our exercise.