Chain rule stepper: outside, inside, and the factor students forget
Year 12 Mathematics Advanced · build y = f(g(x)) from an outside and an inside, step through the rule, then check the answer against a gradient measured from the curve
Outside f
Inside g
Step through the chain rule
outside f is differentiated as a wholeinside u = g(x) is left alone, then its derivative multiplies
1
Name the inside and the outside
2
Differentiate the inside with respect to x
3
Differentiate the outside with respect to u, leaving u alone
4
Multiply: dy/dx = dy/du × du/dx
5
Put the inside back in place of u
Check it against the curve
Graph of the composite function with its tangent at the chosen x, used to measure the gradient numerically.
1
gradient measured from the curve—
chain rule f′(u) · g′(x)—
forgot the inside f′(u) only—
differentiated at x, not u f′(x) · g′(x)—
Move x, then compare. Only one candidate keeps matching wherever you put x.
Pick the outside and inside that make a named function.