Calc2.39

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Find the slope of the tangent line to the graph f(x)=6x when x=3.

Remember, if the derivative exists at a point, it gives us the instantaneous rate of change for a graph that point. Another way to think about that is that it gives us the slope, or constant rate of change, of the line which is tangent to the graph at that point. To find the answer to this problem, find the derivative of f(x) and plug in x=3.

f'(x)=6\,

The derivative of this function is constant, it is always 6. This is because this is a straight line. Its rate of change is constant. If we were to find the slope of the line, we would also get 6. This equation of the line is in the slope-intercept form and we can tell just by looking at the equation that the slope is 6. So,

f'(3)=6\,

This tells us that if we were to draw the line tangent to this graph at the point (3,18), the slope of that line would be 6.


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