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# Differentials

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

## Chain Rule with More Variables

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

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

# Directional Derivatives

## Implications

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

# Lagrange Multipliers

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

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

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

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).