Use the following table to find the given derivatives. ...

Use the following table to find the given derivatives. $$\begin{array}{llllll}x & 1 & 2 & 3 & 4 & 5 \\\hline f(x) & 5 & 4 & 3 & 2 & 1 \\f^{\prime}(x) & 3 & 5 & 2 & 1 & 4 \\g(x) & 4 & 2 & 5 & 3 & 1 \\g^{\prime}(x) & 2 & 4 & 3 & 1 & 5\end{array}$$ $$\left.\frac{d}{d x}\left[\frac{x f(x)}{g(x)}\right]\right|_{x=4}$$

Use the following table to find the given derivatives.
$$\begin{array}{llllll}x & 1 & 2 & 3 & 4 & 5 \\\hline f(x) & 5 & 4 & 3 & 2 & 1 \\f^{\prime}(x) & 3 & 5 & 2 & 1 & 4 \\g(x) & 4 & 2 & 5 & 3 & 1 \\g^{\prime}(x) & 2 & 4 & 3 & 1 & 5\end{array}$$
$$\left.\frac{d}{d x}\left[\frac{x f(x)}{g(x)}\right]\right|_{x=4}$$

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Key Concepts

Quotient Rule

The quotient rule is a fundamental technique for differentiating a function that is the ratio of two differentiable functions. It states that if you have a function h(x) = u(x)/v(x), then the derivative h'(x) is given by (u'(x)v(x) - u(x)v'(x)) divided by [v(x)]^2. This rule is essential when differentiating expressions where the numerator and denominator are both functions of x.

Product Rule

The product rule is used when differentiating the product of two functions. It states that if u(x) and v(x) are differentiable, then the derivative of their product is u'(x)v(x) + u(x)v'(x). In the context of differentiating a quotient where the numerator itself involves a product, the product rule is applied to correctly compute the derivative of that part.

Evaluating Derivatives from a Data Table

In problems where function values and their corresponding derivative values are provided in a table, knowing how to extract and substitute these values into differentiation formulas is crucial. This skill allows one to compute the necessary derivative at a specified point without needing the explicit functional form.

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