In Exercises 81 and 82, use the graphs of f and g . Let...

In Exercises 81 and $82,$ use the graphs of $f$ and $g .$ Let $p(x)=f(x) g(x)$ and $q(x)=f(x) / g(x) .$
$$
\begin{array}{l}{\text { (a) Find } p^{\prime}(4)} \\ {\text { (b) Find } q^{\prime}(7)}\end{array}
$$

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Key Concepts

Product Rule

The product rule is a fundamental rule in differential calculus used to find the derivative of a product of two functions. It states that if p(x) = f(x)g(x), then the derivative p'(x) is given by f'(x)g(x) + f(x)g'(x). This concept is essential when differentiating functions that are products of other functions whose derivatives are known or can be determined.

Quotient Rule

The quotient rule is used to differentiate a function that is the quotient of two other functions. Specifically, if q(x) = f(x)/g(x), then its derivative is given by [f'(x)g(x) - f(x)g'(x)] divided by [g(x)]^2. This rule is crucial for handling differentiations where one function is divided by another, and it helps in simplifying and understanding the behavior of the resulting derivative.

Graphical Interpretation of Derivatives

Interpreting the derivatives from graphs involves understanding that the slope of the tangent line at any given point on a function's graph represents its derivative at that point. When functions and their slopes are provided graphically, one can use these visual cues to evaluate the necessary function values and their corresponding derivatives, which are then applied in rules such as the product rule and quotient rule.

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