Geometric Interpretation of Diﬀerentiation y Q Secant line Tangent line P f(x) x0 x0+∆x Figure 1: A graph with secant and tangent lines. The derivative of f (x) at x = x0 is the slope of the tangent line to the graph of f (x) at the point (x0 , f (x0 )). But what is a tangent line? • It is NOT just a line that meets the graph at one point. • It is the limit of the secant lines joining points P = (x0 , f (x0 )) and Q on the graph of f (x) as Q approaches P . The tangent line touches the graph at (x0 , f (x0 )); the slope of the tangent line matches the direction of the graph at that point. The tangent line is the straight line that best approximates the graph at that point. Given a graph of our function, it’s not hard for us to draw the tangent line to the graph. However, we’ll want to do computations involving the tangent line and so will need a computational method of ﬁnding the tangent line. How do we compute the equation of the line tangent to the graph of the function f (x) at a point P = (x0 , y0 )? We know that the equation of the straight line with slope m through the point (x0 , y0 ) is y − y0 = m(x − x0 ), so in the abstract we know the equation of the tangent line. To get a speciﬁc equation for the line, we’ll need to know the coordinates x0 and y0 of the point P . If we know x0 we can ﬁnd y0 = f (x0 ) by substituting the value x0 in to the expression for f (x). The second thing we need to know is the slope, m = f � (x0 ), which we call the derivative of f . Deﬁnition: The derivative f � (x0 ) of f at x0 is the slope of the tangent line to y = f (x) at the point P = (x0 , f (x0 ). 1