Computing dz/dt for z=x²y where x=cos t, y=sin t

Calculus 3 · Axiom Academy

EXAMPLE Computing for z = x^2 y , , Applying the multivariable chain rule when both variables depend on one parameter Let z = x^2 y , where and . Use the multivariable chain rule to find as a function of t . Nice work. You applied the multivariable chain rule from start to finish and simplified with a trig identity. Here is what carried the solution: The chain rule: When z = f(x, y) and both x and y depend on t , use . Build each piece separately: Find and , then and . Mind the signs: is where the negative enters — losing it is the classic mistake. Simplify with an identity: collapses the answer to , equivalently . Check by substitution: Put x, y into z first to get ; differentiating directly gives the same result. This same technique powers related-rates problems and optimization in higher dimensions, and it extends directly to functions of more variables.

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