Prove formulas for the partial derivatives of x^y

by RoRi

November 17, 2015

Define a function of two variables, with $x> 0$ ,

$f(x,y) = x^y.$

Prove

$\frac{\partial f}{\partial x} = yx^{y-1} \qquad \text{and} \qquad \frac{\partial f}{\partial y} = x^y \log x.$

Proof. First, we write,

$f(x,y) = x^y = e^{ y \log x}.$

Then we compute the derivatives using the chain rule and formulas for the derivatives of the exponential and logarithm,

$\frac{\partial f}{\partial x} = e^{y \log x} \cdot \frac{y}{x} = yx^{y-1}.$

And,

$\frac{\partial f}{\partial y} = e^{y \log x} \cdot \log x = x^y \log x. \qquad \blacksquare$

Stumbling Robot