Exercise 1079Calculus › Multivariable calculus › Partial derivativesL1Added 3 Nov 2026

General chain rule

#chain rule#composition#multivariable

Statement

Let $f(u,v)=u^2v$ and $u(x,y)=x+y$ , $v(x,y)=xy$ . Compute $\frac{\partial}{\partial x}f(u,v)$ and $\frac{\partial}{\partial y}f(u,v)$ .

Hint

Apply the chain rule:
$\frac{\partial f}{\partial x}=f_u u_x+f_v v_x.$

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Partial derivatives of $f$ . $f_u=2uv$ , $f_v=u^2$ .

Chain rule for $\partial_x$ .
$\frac{\partial f}{\partial x}=2uv\cdot1+u^2\cdot y=2(x+y)(xy)+y(x+y)^2=(x+y)(2xy+y(x+y)).$

Chain rule for $\partial_y$ .
$\frac{\partial f}{\partial y}=2uv\cdot1+u^2\cdot x=2(x+y)(xy)+x(x+y)^2=(x+y)(2xy+x(x+y)).$

Conclusion. Both partial derivatives are polynomials in $x,y$ .