Use the following table to find the given derivatives. x 1 2 3 4 f(x)

step 1 Let's find the derivative of this function, evaluated at x equals 4, using the table in the book

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

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

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

Key Concepts

Quotient Rule

The quotient rule is used for finding the derivative of a function that is the quotient of two differentiable functions. It states that if u(x) and v(x) are functions with v(x) ? 0, then the derivative of the quotient u(x)/v(x) is (u'(x)v(x) - u(x)v'(x)) divided by [v(x)]². This rule is crucial when dealing with ratios of functions, as it provides a systematic way to incorporate both the numerator's and denominator’s contributions to the rate of change.

Evaluation of Derivatives Using Tabulated Data

Often in calculus problems, values of functions and their derivatives are provided in a table at specific points. In such cases, it is common to directly substitute these known values into the derivative formulas, which have been derived using rules like the product or quotient rule. This approach is particularly useful when deriving expressions at a specific point, enabling the solver to compute the exact numerical value of the derivative based on pre-calculated information.

Product Rule

The product rule is a fundamental technique in differentiation that enables one to compute the derivative of the product of two functions. It states that if u(x) and v(x) are differentiable functions, then the derivative of their product is given by u'(x)v(x) + u(x)v'(x). This rule is essential when functions that multiply together need to be differentiated, particularly in problems combining multiple factors, as it breaks down a complex differentiation task into simpler parts.

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