Questions: The equation of the tangent line to the graph of a function (f) at ((2,6)) is (y=-3 x+16). What is (f'(2))?

Transcript text: The equation of the tangent line to the graph of a function $f$ at $(2,6)$ is $y=-3 x+16$. What is $f^{\prime}(2)$ ?

Solution

Solution Steps

Step 1: Understand the relationship between the tangent line and the derivative

The slope of the tangent line to the graph of a function $ f $ at a point $ (a, f(a)) $ is equal to the derivative of the function at that point, $ f^{\prime}(a) $.

Step 2: Identify the slope of the given tangent line

The equation of the tangent line is $ y = -3x + 16 $. The slope of this line is $ -3 $.

Step 3: Relate the slope to the derivative

Since the slope of the tangent line at $ (2, 6) $ is $ -3 $, the derivative of $ f $ at $ x = 2 $ is also $ -3 $. Therefore: \[ f^{\prime}(2) = -3 \]

Final Answer

$\boxed{f^{\prime}(2) = -3}$

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