Unit 6Differentiation1.3Gradients of the graph ofy=x2The graph ofy=x2has a gradient ateverypoint, because it has atangent, which is not vertical, at every point.Let’s try to find the gradient of this graph at the point (1,1), which isshown in Figure 10. That is, we want to find the gradient of the tangent tothe graph at this point, which is also shown in Figure 10.xy¡3¡2¡1123¡11234Figure 10The point (1,1) on the graph ofy=x2and the tangent atthis pointThe way to find the gradient at (1,1) is to begin by thinking about howyou could find anapproximatevalue for this gradient. Here’s how you cando that. You choose a second point on the graph ofy=x2, fairly close to(1,1), as illustrated in blue in Figure 11. The straight line that passesthrough both (1,1) and the second point is an approximation for thetangent to the graph at (1,1). So the gradient of this line, which you cancalculate using the two points on the line in the usual way, is anapproximation for the gradient of the graph at (1,1).xy¡3¡2¡1123¡11234Figure 11An approximation to the tangent to the graph ofy=x2at(1,1)The closer to (1,1) you choose the second point to be, the better theapproximation will be. For example, the second point in Figure 12 willgive a better approximation than the second point in Figure 11.218