The Difference Quotient

Pre-Calculus · Axiom Academy

LESSON The Difference Quotient One formula, one picture: the slope of a secant line — and how shrinking it into a tangent gives the instantaneous rate of change. Pick an input x and step a little distance h to the right, landing at x+h . Join the two points on the curve, and , with a straight line — a secant line . The difference quotient is just that line's slope: rise over run. Now slide the second point toward the first by letting h get smaller. The secant pivots, and as it settles onto the tangent line — the line that just grazes the curve at a single point. Its slope is the instantaneous rate of change . The two points are far apart. The secant is a rough, average rate of change over a wide interval. The far point slides inward. The secant tilts toward the curve's local direction at x . The secant becomes the tangent. Its slope is the rate of change at the single point x . That limiting slope is exactly what calculus calls the derivative f'(x) . For f(x)=x^2 at x=1 , the secant slope is 2x+h . As h runs , the readout goes , closing in on 2 — the tangent slope at x=1 . The payoff move is algebraic: build the difference quotient, then cancel the h in the denominator so it survives sending . Here is the whole computation for f(x)=x^2 . — you cannot just set h=0 ; that is . 2x+h — now h=0 is safe and gives 2x , the instantaneous rate. You've seen the difference quotient as the slope of a secant line, watched it collapse into a tangent as , and computed it for f(x)=x^2 .

This is the written version of the interactive lesson above. See the full Pre-Calculus course.