Partial derivatives

Partial derivatives

$f_x$ is the derivative with respect to $x$ treating $y$ as a constant; $f_y$ treats $x$ as a constant.

Second partial derivatives $f_{xx}$, $f_{xy}$, $f_{yy}$ are found the same way. If $f_{xy}$ and $f_{yx}$ are continuous, then $f_{xy} = f_{yx}$ (Schwarz's theorem).

Total differential and tangent planes

The approximation $f(a + h,\ b + k) \approx f(a,\ b) + f_x(a,\ b)h + f_y(a,\ b)k$ lets you estimate values such as $\sqrt{3.01^2 + 3.99^2}$ by hand.

The chain rule

Extrema and constrained extrema

At a critical point ($f_x = f_y = 0$), compute $D = f_{xx}f_{yy} - f_{xy}^2$:

• $D > 0$ and $f_{xx} > 0$: local minimum
• $D > 0$ and $f_{xx} < 0$: local maximum
• $D < 0$: not an extremum (saddle point)

For extrema subject to $g(x,\ y) = 0$, use Lagrange multipliers: with $F = f - \lambda g$, solve $F_x = F_y = 0$ and $g = 0$ to find the candidates.