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# Differentials

• $df = f_xdx + f_ydy+f_zdz$ο»Ώ
• $df \neq \Delta f$ο»Ώ
• $\Delta f = f_x\Delta x + f_y\Delta y + f_z\Delta z$ο»Ώ
• $df$ο»Ώ is used to encode infinitesimal changes
• used to act as a placegolder value
• divide wrt time to get rate of change $\rightarrow$ο»Ώ CHAIN RULE

## Chain Rule with More Variables

Let $w = f(x, y)$ο»Ώ when $x(u, v), y(u, v)$ο»Ώ then,

Note: $\vec{\nabla}w \ \perp \text{ level surfaces}$ο»Ώ (tangent to the level surface at any given point)

# Directional Derivatives

## Implications

Direction of $\vec{\nabla}w$ο»Ώ is the direction of fastest increase of $w$ο»Ώ

# Lagrange Multipliers

Goal: minima/maximize a multi-variable function ($min/max\ \ f(x, y, z)$ο»Ώ) where $x, y, z$ο»Ώ are not independent and $\exists$ο»Ώ $g(x, y, z) = c$ο»Ώ.

These can be obtained on combining the given restraints with the following.

Basic idea: to find $(x, y)$ο»Ώ where the level curves of $f$ο»Ώ and $g$ο»Ώ are tangent to each other ($\vec{\nabla}f \parallel \vec{\nabla}g$ο»Ώ).

Note: Take care that the point is indeed a maxima or minima as required and not just a saddle point (second derivative test won't be applicable so be vigilant).