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 $h^{\prime}(2)$ if $h(x)=\frac{f(x)}{g(x)}$

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 $h^{\prime}(2)$ if $h(x)=\frac{f(x)}{g(x)}$

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Key Concepts

Differentiability

Differentiability is a fundamental property of a function that implies the function has a derivative at a given point. This concept ensures that small changes in the input result in predictable changes in the output and is essential for applying derivative rules, such as the quotient rule used in this problem.

Quotient Rule

The quotient rule is a method for differentiating functions that are expressed as a ratio of two differentiable functions. It provides a systematic way to compute the derivative by taking the derivative of the numerator and the denominator, combining them in a specific formula that accounts for the interaction between the two functions.

Evaluating Derivatives Using Given Data

Evaluating derivatives using provided tables or specific values involves substituting the known function and derivative values at a specific point into the appropriate differentiation formula. This process is critical in problems where only discrete data is available, ensuring that the calculation of the derivative at a point is accurate using the information given.

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