Consider the parabola y = 5x − x2. (a) find the slope m of the tangent line to the parabola at the point (1, 4).

The tangent line to a curve is the one that coincides with the curve at a point and with the same derivative, that is, the same degree of variation.
 We have then:
 y = 5x-x²
 Deriving:
 y '= 5-2x
 In point (1, 4)
 The slope is:
 y (1) '= 5-2 * (1)
 y (1) '= 3
 The equation of the line will be:
 y-f (a) = f '(a) (x-a)
 We have then:
 y-4 = 3 (x-1)
 Rewriting:
 y = 3x-3 + 4
 y = 3x + 1
 
 the tangent line to the parabola at the point (1, 4) is
 y = 3x + 1
 the slope m is
 m = 3

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