Rates in Specific Directions

Calculus 3 · Axiom Academy

Standing on a hillside, "how steep is it?" depends on which way you face. Pick a direction — the directional derivative tells you the slope that way. One point, one function — but a different slope every way you turn On a curve y=f(x) there is just one slope at each point. On a surface z=f(x,y) you are standing on a hillside: face one way and it climbs, turn 90° and it might be dead level. So the honest question isn't "what is the slope?" — it's "what is the slope in this direction ?" That number is the directional derivative . Here is a tilted field f(x,y)=3x+4y seen from above as a contour map — each line joins points of equal height. At the point P the arrow points straight uphill. Watch a unit direction swing all the way around: the slope rises to a peak when lines up with , falls to zero when runs along a contour, and goes negative pointing downhill. The slope in direction is just how much of points along — the shadow it casts on the uphill arrow. Turn the dial and read the slope in any direction Drag the handle to aim the unit vector wherever you like. The readout is , where is the angle between and . Find the two directions that make it biggest and zero — and notice that pointing along +x or +y just recovers the partial derivatives f_x and f_y . Biggest uphill, zero along a contour, most-negative downhill — one function, a whole fan of slopes.

This is the written version of the interactive lesson above. See the full Calculus 3 course.