For the following exercises, assume that f(x) and g(x) are...

For the following exercises, assume that $f(x)$ and $g(x)$ are both differentiable functions with values as given in the following table. Use the following table to calculate the following derivatives. $$\begin{array}{|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} \\ \hline f(x) & {3} & {5} & {-2} & {0} \\ \hline g(x) & {2} & {3} & {-4} & {6} \\ \hline f^{\prime}(x) & {-1} & {7} & {8} & {-3} \\ \hline g^{\prime}(x) & {4} & {1} & {2} & {9} \\ \hline\end{array}$$ Find $h^{\prime}(4)$ if $h(x)=\frac{1}{x}+\frac{g(x)}{f(x)}$.

For the following exercises, assume that $f(x)$ and $g(x)$ are both differentiable functions with values as given in the following table. Use the following table to calculate the following derivatives.
$$\begin{array}{|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} \\ \hline f(x) & {3} & {5} & {-2} & {0} \\ \hline g(x) & {2} & {3} & {-4} & {6} \\ \hline f^{\prime}(x) & {-1} & {7} & {8} & {-3} \\ \hline g^{\prime}(x) & {4} & {1} & {2} & {9} \\ \hline\end{array}$$
Find $h^{\prime}(4)$ if $h(x)=\frac{1}{x}+\frac{g(x)}{f(x)}$.

Key Concepts

Quotient Rule

The quotient rule is used when differentiating a function that is the ratio of two differentiable functions. It states that if a function h(x) is defined as g(x)/f(x), then its derivative h'(x) is given by (g'(x)f(x) - g(x)f'(x))/(f(x))^2. This rule is critical when the numerator and denominator are both functions of x and need to be differentiated separately before combining them.

Evaluating Derivatives Using Given Values

In problems where function values and their derivatives at specific points are provided, the process of evaluation involves substituting these values directly into the derivative formulas (such as those obtained from the reciprocal and quotient rules). This technique facilitates the computation of the derivative at a particular point, making use of the supplied numerical data.

Differentiation Rules

Differentiation rules are the foundational formulas used to calculate the derivative of a function. They include the sum rule, product rule, quotient rule, and chain rule, each designed to handle different configurations of functions. These rules are essential in determining how functions change at any given point and serve as the building blocks for more complex derivative calculations.

Reciprocal Function Differentiation

Differentiating a reciprocal function, such as 1/x, involves applying the power rule by rewriting the function as x^(-1). The derivative is then found using the power rule, yielding -1/x^2. This technique is fundamental when dealing with functions expressed as the reciprocal of a variable.

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