Average and instantaneous gradients

The average gradient is delta ydelta x; the derivative is its limiting value as delta x 0. A secant joins two points. Moving the second point towards the first makes its slope approach the tangent slope. Method Find the two output values; divide their difference by the signed input change; compare with the tangent gradient. Example y=x2, x:1 1.1, delta y=0.21, delta ydelta x=2.1. The tangent gradient at x=1 is 2. Exam check Do not call the finite average the derivative. The derivative is the limiting value, not an instruction to substitute zero into a zero denominator.