Exercise 523Calculus › Multivariable calculus › Partial derivativesL1Added 3 Nov 2026

Euler's theorem for homogeneous functions

#homogeneous functions#Euler#partial derivatives

Statement

Show that if $f$ is homogeneous of degree $k$ (i.e. $f(tx,ty)=t^kf(x,y)$ ), then
$x f_x+y f_y=k f.$

Hint

Differentiate the homogeneity identity with respect to $t$ and evaluate at $t=1$ .

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Identity. $f(tx,ty)=t^kf(x,y)$ .

Differentiate. $\frac{d}{dt}$ : $x f_x(tx,ty)+y f_y(tx,ty)=k t^{k-1}f(x,y)$ .

At $t=1$ . $xf_x(x,y)+yf_y(x,y)=kf(x,y)$ .