Scribd
Upload
23 views18 pages

Section 145

Chapter 14 covers Partial Derivatives, focusing on functions of several variables, limits, continuity, and the chain rule. It includes examples illustrating the chain rule in different cases and applications of implicit differentiation. Key concepts such as tangent planes, directional derivatives, and Lagrange multipliers are also discussed.

Uploaded by

alphacalvin81
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
23 views18 pages

Section 145

Chapter 14 covers Partial Derivatives, focusing on functions of several variables, limits, continuity, and the chain rule. It includes examples illustrating the chain rule in different cases and applications of implicit differentiation. Key concepts such as tangent planes, directional derivatives, and Lagrange multipliers are also discussed.

Uploaded by

alphacalvin81
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 18

1

Chapter 14 : Partial Derivatives


2

Chapter 14 : Partial Derivatives

14.1 Functions of Several Variables

14.2 Limits and Continuity

14.3 Partial Derivatives

14.4 Tangent Planes and Linear Approximations

14.5 The Chain Rule

14.6 Directional Derivatives and The Gradient Vector

14.7 Maximum and Minimum Values

14.8 Lagrange Multipliers


3

14.5 The Chain Rule

Objectives :

 Learn the chain rule for functions of several variables


4

Chain Rule : Case 1


5

Example 1

If z = x2 y + 3xy 4 where x = sin 2t and y = cos t.

Find dz/dt when t = 0.


6

Chain Rule : Case 2


7

Example 2

∂z ∂z
If z = ex sin y where x = st2 and y = s2 t. Find ∂s and ∂t
8

Chain Rule : General Version


9

Example 3

If u = x4 y + y 2 z 3 where

x = rset , y = rs2 e−t , z = r2 s sin t


∂u
find the value of , when r = 2, s = 1, t = 0.
∂s
10
11
12

Example 4

If g(t, s) = f (s2 − t2 , t2 − s2 ) and f is differentiable.

Show that g satisfies the equation

∂g ∂g
t +s =0
∂s ∂t
13

Example 5

If z = f (x, y) has continuous second-order partial derivatives

2 2 ∂z ∂2z
and x = r + s , y = 2rs. Find and
∂r ∂r2
14

Implicit Differentiation

If x and y are related implicitly by F (x, y) = 0.

If F is differentiable, we apply Case 1 of the Chain rule


15

Example 6

Find y 0 if x3 + y 3 = 6xy
16

Implicit Differentiation

If z is given implicitly as a function of x and y as F (x, y, z) = 0.

If F is differentiable and z is differentiable with respect to x


and y, we apply the Chain rule
17

Example 7

∂z ∂z
Find ∂x and ∂y if x3 + y 3 + z 3 + 6xyz = 1
18

Keywords to remember

 The Chain Rule


 Implicit Differentiation

You might also like