Let F(x) = f(x) + g(x),G(x) = f(x) - g(x), and H(x) = 3f(x) + 2g(x), where the graphs of f and g are shown in the figure. Find each of the following.
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H'(2)

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Step 1: Understand the problem. We are given three functions: F(x) = f(x) + g(x), G(x) = f(x) - g(x), and H(x) = 3f(x) + 2g(x). We need to find the derivative of H at x = 2, denoted as H'(2).

Step 2: Use the linearity of differentiation. The derivative of a sum is the sum of the derivatives, and the derivative of a constant times a function is the constant times the derivative of the function. Therefore, H'(x) = 3f'(x) + 2g'(x).

Step 3: Evaluate the derivatives of f and g at x = 2. From the graph, determine the values of f'(2) and g'(2). These are the slopes of the tangent lines to the graphs of f and g at x = 2.

Step 4: Substitute the values of f'(2) and g'(2) into the expression for H'(x). This gives H'(2) = 3f'(2) + 2g'(2).

Step 5: Calculate H'(2) using the values obtained from the graph. This will give you the rate of change of H at x = 2.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. It is a fundamental concept in calculus, representing the slope of the tangent line to the graph of the function at a given point. For a function F(x), the derivative F'(x) can be found using rules such as the sum, difference, and constant multiple rules.

Sum and Difference of Functions

When dealing with the sum or difference of two functions, the derivative can be computed by applying the sum and difference rules. Specifically, if F(x) = f(x) + g(x), then F'(x) = f'(x) + g'(x). Similarly, for G(x) = f(x) - g(x), G'(x) = f'(x) - g'(x). This property allows for the straightforward calculation of derivatives for combined functions.

Linear Combination of Functions

A linear combination of functions involves multiplying each function by a constant and then adding the results. In the case of H(x) = 3f(x) + 2g(x), the coefficients 3 and 2 indicate how much each function contributes to H. The derivative of H can be found by applying the constant multiple rule, which states that the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.

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