Use the given graphs of f and g to find each derivative. <IMAGE>
c. d/dx ((f(x) / g(x)) |x=3

Verified step by step guidance

1. To find the derivative of the function \( \frac{f(x)}{g(x)} \) at \( x = 3 \), we will use the Quotient Rule. The Quotient Rule states that if you have a function \( h(x) = \frac{f(x)}{g(x)} \), then its derivative \( h'(x) \) is given by \( h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} \).
2. First, identify \( f(x) \) and \( g(x) \) from the graphs at \( x = 3 \). Determine the values of \( f(3) \) and \( g(3) \) from the graph.
3. Next, find the derivatives \( f'(x) \) and \( g'(x) \) at \( x = 3 \). This involves determining the slopes of the tangent lines to the graphs of \( f \) and \( g \) at \( x = 3 \).
4. Substitute the values \( f(3) \), \( g(3) \), \( f'(3) \), and \( g'(3) \) into the Quotient Rule formula: \( h'(3) = \frac{f'(3)g(3) - f(3)g'(3)}{(g(3))^2} \).
5. Simplify the expression obtained from the Quotient Rule to find the derivative of \( \frac{f(x)}{g(x)} \) at \( x = 3 \).

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. Derivatives are fundamental in calculus for understanding rates of change and are denoted as f'(x) or df/dx.

Quotient Rule

The Quotient Rule is a formula used to find the derivative of a function that is the quotient of two other functions. If h(x) = f(x) / g(x), the derivative h'(x) is given by h'(x) = (f'(x)g(x) - f(x)g'(x)) / (g(x))^2. This rule is essential when differentiating functions that are divided by one another.

Evaluating Derivatives at a Point

Evaluating a derivative at a specific point involves substituting the x-value into the derivative function. For example, to find d/dx (f(x) / g(x)) at x=3, you first apply the Quotient Rule to find the derivative and then substitute x=3 into the resulting expression. This process provides the instantaneous rate of change of the function at that particular point.