Chain Rule, inside function is linear (v1d)
- Dick Lane
Right panel shows y = f(x). It is composite of function u=h(x) followed by y = g(u); thus f(x) = g( h(x) ). Function h is defined in right panel as h(x) = p*(x - q) where parameters p & q have values controlled by sliders. Function g is defined in left panel with expression [in terms of u] entered into an input box. The graph of g is shown in the left panel. Point A=(a,0) on x-axis in right panel can be moved; the line tangent to f at point (a,f(a)) is displayed together with its slope, f'(a). Number b=h(a) is computed, point B=(b,0) is shown in the left panel and line tangent to g at u=b is shown; its slope, g'(b), is also shown.
Explorations: 1) set p=1 and vary q; this will involve horizontal shifts 2) set q=0 and vary p between 1 and 3; this involves change of horizontal scale 3) keep q=0 and vary p between 0.1 and 1; this also involves horizontal scale 4) q=0 and p=-1; graph of g is reflected about y-axis 5) try slightly more complicated changes involving both p and q