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The Derivative

Problems

Limit Definition of the Derivative

The Derivative Function

Problem : Calculate the derivative of f (x) = x 2 at x = 1 .

Substituting 1 for x 0 in the formula for the derivative, we have


f'(1) =  
  =  
  = 2 + Δx  
  = 2  

Observing the graph of f (x) = x 2 , we see that this is the slope of the tangent to the graph at the point (1, f (1)) = (1, 1) .

Problem : Find the vertex of the parabola f (x) = x 2 + 2x + 2 using the derivative.

At the vertex, the tangent line to the graph will be horizontal, with slope 0 . Therefore, we search for an x such that f'(x) = 0 . We have


f'(x) = limΔx→0  
  = limΔx→0  
  = limΔx→0  
  = limΔx→02x + 2 + Δx  
  = 2(x + 1)  

Thus f'(x) = 0 if and only if x = - 1 . Now f (x) = (- 1)2 + 2(- 1) + 2 = 1 , so the vertex of the parabola is (- 1, 1) . We can check this by noting that f (x) - 1 = (x + 1)2 , so the graph of f (x) is the graph of x 2 translated 1 unit to the left and 1 unit up.

Problem : Find the equation of the tangent line to the graph of f (x) = x 3 at x = 2 .

First we compute f'(2) :


f'(2) = limΔx→0  
  = limΔx→0  
  = 12  

The equation of the line through (2, f (2)) = (2, 8) with slope 12 is given by y = 12(x - 2) + 8 .

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