Differentials and Tangent Lines

Suppose we are given a curve y = f(x) and at a point (a, b) on the curve the slope f'(a) is defined. Then the tangent line to the curve at the point (a, b), illustrated in Figure 2.2.1, is defined to be the straight line which passes through the point (a, b) and has the same slope as the curve at x = a. Thus the tangent line is given by the equation

l(x) - b = f'(a)(x - a),

or

l(x) = f'(a)(x - a) + b.