Derivatives from a graph Let F = f + g and G = 3f - g, where the graphs of f and g are shown in the figure. Find the following derivatives.
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G'(2)

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the derivative of the function G at x = 2, where G is defined as G = 3f - g.
2. Step 2: Use the linearity of derivatives. The derivative of G, G'(x), can be found using the rule: G'(x) = (3f - g)' = 3f'(x) - g'(x).
3. Step 3: Evaluate the derivatives of f and g at x = 2. From the graph, determine the slopes of the tangent lines to the curves of f and g at x = 2, which represent f'(2) and g'(2) respectively.
4. Step 4: Substitute the values of f'(2) and g'(2) into the expression for G'(2). This gives G'(2) = 3f'(2) - g'(2).
5. Step 5: Calculate G'(2) using the values obtained from the graph. This will give you the rate of change of G at x = 2.

Key Concepts

Derivative

A derivative represents the rate of change of a function with respect to its variable. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In graphical terms, the derivative at a point corresponds to the slope of the tangent line to the curve at that point.

Sum and Difference of Functions

When dealing with the derivatives of functions that are added or subtracted, the derivative of the sum or difference is simply the sum or difference of their derivatives. This is expressed mathematically as (f + g)' = f' + g' and (3f - g)' = 3f' - g'. This property allows for straightforward computation of derivatives for combined functions.

Evaluating Derivatives at Specific Points

To find the derivative at a specific point, such as G'(2), one must first compute the derivative function and then substitute the given value into this function. This process often involves using the values of the original functions and their derivatives at that point, which can be obtained from the graph or through analytical methods.