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Differentiability

Those friendly functions that don't contain breaks, bends or cusps are "differentiable". Take their derivative, or just infer some facts about them from the Mean Value Theorem.

Slope of a Curve

         

Graph of y=2x+3

Graph of y=2x+3

Sally drew the line \( y=2x+3 \) on the board. What is the slope of this line?

What is the slope of the line tangent to the curve \( y = \ln x + 20x \) at \( x = \frac{1}{7} \) ?

What is the slope of the line tangent to the graph of the function \( f(x) = - \frac{4}{x} \) at \( x = \frac{1}{9} \) ?

What is the slope of the line tangent to the graph of the function \( f(x) = 6 \cos x + 6 \sin x \) at \( x = 20 \pi \) ?

What is the slope of the tangent line of the curve \( f(x) = x^2 + 2013 \) at \( x = 29 \)?

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