Transcript Slide 1

Chapter 14 – Partial Derivatives
14.5 The Chain Rule
Objectives:
 How to use the Chain Rule and
applying it to applications
 How to use the Chain Rule for
Implicit Differentiation
Dr. Erickson
14.5 The Chain Rule
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Chain Rule: Single Variable
Functions


Recall that the Chain Rule for functions of a single
variable gives the following rule for differentiating a
composite function.
If y = f (x) and x = g (t), where f and g are differentiable
functions, then y is indirectly a differentiable function of
t, and
dy dy dx

dt dx dt
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14.5 The Chain Rule
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Chain Rule: Multivariable Functions


For functions of more than one variable, the Chain Rule
has several versions.
◦ Each gives a rule for differentiating
a composite function.
The first version (Theorem 2) deals with
the case where z = f (x, y) and each of
the variables x and y is, in turn, a function
of a variable t.
◦ This means that z is indirectly a function of t,
z = f (g(t), h(t)), and the Chain Rule gives a formula
for differentiating z as a function of t.
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Chain Rule: Case 1

Since we often write ∂z/∂x in place of ∂f/∂x, we
can rewrite the Chain Rule in the form
dz z dx z dy


dt x dt y dt
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Example 1 – pg. 930 # 2

Use the chain rule to find dz/dt or dw/dt.
z  cosx  4 y , x  5t ,
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y  1/ t
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Example 2 – pg. 930 # 6

Use the chain rule to find dz/dt or dw/dt.
w  ln x 2  y 2  z 2 , x  sin t ,
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y  cos t , z  tan t
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Chain Rule: Case 2

Case 2 of the Chain Rule contains three types of
variables:
◦ s and t are independent variables.
◦ x and y are called intermediate variables.
◦ z is the dependent variable.
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Using a Tree Diagram with Chain
Rule

We draw branches from the dependent variable z to the
intermediate variables x and y to indicate that z is a
function of x and y.
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Tree Diagram

Then, we draw branches from x and y to the independent
variables s and t.
◦ On each branch,
we write the
corresponding
partial derivative.
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Tree Diagram

To find ∂z/∂s, we find the product of the partial
derivatives along each path from z to s and then add
these products:
z z x z y


s x s y s
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Example 3 – pg. 930 # 12

Use the Chain rule to find ∂z/∂s and ∂z/∂t.
z  tan(u / v), u  2s  3t , v  3s  2t
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Chain Rule: General Version
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Example 4

Use the Chain Rule to find the indicated
partial derivatives.


R  ln u 2  v 2  w2 , u  x  2 y, v  2 x  y, w  2 xy;
R R
,
, when x  y  1
x
y
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Example 5

Use the Chain Rule to find the indicated
partial derivatives.
Y  w tan1 uv, u  r  s, v  s  t , w  t  r;
Y
Y
Y
,
,
when r  t  1, s  0
r
s
t
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Implicit Differentiation

The Chain Rule can be used to give
a more complete description of the
process of implicit differentiation that was
introduced in Sections 3.5 and 14.3
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Implicit Differentiation

If F is differentiable, we can apply Case 1 of the Chain
Rule to differentiate both sides of the equation
F(x, y) = 0 with respect to x.
◦ Since both x and y are functions of x,
we obtain:
F dx F dy

0
x dx y dx
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Implicit Differentiation
However, dx/dx = 1.
 So, if ∂F/∂y ≠ 0, we solve for dy/dx
and obtain:

F
Fx
dy

x


F
dx
Fy
y
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Implicit Differentiation


Now, we suppose that z is given implicitly as a function
z = f(x, y) by an equation of the form F(x, y, z) = 0.
◦ This means that F(x, y, f(x, y)) = 0
for all (x, y) in the domain of f.
If F and f are differentiable, then we can use the Chain
Rule to differentiate the equation
F(x, y, z) = 0 as
follows:
F x F y F z


0
x x y x z x
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Implicit Differentiation

However,


( x)  1 and
( y)  0
x
x
So, that equation becomes:
F F z

0
x z x
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Implicit Differentiation

If ∂F/∂z ≠ 0, we solve for ∂z/∂x and obtain
the first formula in these equations.
F
z
  x
F
x
z
F
z
y

F
y
z
◦ The formula for ∂z/∂y is obtained in a similar
manner.
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Example 6

Use equation 7 to find ∂z/∂x and ∂z/∂y .
yz  lnx  z 
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More Examples
The video examples below are from section
14.5 in your textbook. Please watch them
on your own time for extra instruction.
Each video is about 2 minutes in length.
◦ Example 2
◦ Example 4
◦ Example 5
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