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Chapter 14 – Partial Derivatives14.3 Partial Derivatives
14.3 Partial Derivatives
Objectives: Understand the various aspects
of partial derivatives
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Partial Derivative w.r.t. x at (a, b)
In general, if f is a function of two variables x and y, suppose we let only x vary while keeping y fixed, say y = b, where b is a constant.
Then, we are really considering a function of a single variable x
g(x) = f(x, b)
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Partial Derivative w.r.t. x at (a, b)
If g has a derivative at a, we call it the partial derivative of f with respect to x at (a, b).
We denote it by:
fx(a, b)
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Partial Derivative w.r.t. x at (a, b)
So we have,
By using the definition of derivative, this equation becomes
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Partial Derivative w.r.t. y at (a, b)
Similarly, the partial derivative of f with respect to y at (a, b), denoted by fy(a, b), is obtained by:
◦Keeping x fixed (x = a)
◦Finding the ordinary derivative at b of the function G(y) = f(a, y)
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Partial Derivative w.r.t. y at (a, b)
So we have,
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Definition - Partial DerivativesIf we now let the point (a, b) vary
in Equations 2 and 3, fx and fy become functions of two variables.
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Notation for Partial DerivativesIf z = f (x,y), we can write
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Rule for finding Partial Derivatives
z = f (x,y)
To find fx, regard y as a constant and differentiate f (x,y) w.r.t. x.
To find fy, regard x as a constant and differentiate f (x,y) w.r.t. y.
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Example 1 – pg. 912 # 16Find the first partial derivatives of the
function.4 3 2( , ) 8f x y x y x y
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Example 2Find the first partial derivatives of the
function.
( , ) arctanf x t x t
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Function of more than Two Variables
A function of three variables has the partial derivative w.r.t. x is defined as
and is found by treating y and z as constants and differentiating the function w.r.t. x
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Example 3Find the first partial derivatives of the
function.
( ) sinf xyz x y z
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Example 4Find the first partial derivatives of the
function.2
( , , , )2
xyf x y z t
t z
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Higher DerivativesIf f is a function of two variables, then its
partial derivatives fx and fy are also functions of two variables.
So, we can consider their partial derivatives
(fx)x , (fx)y , (fy)x , (fy)y
These are called the second partial derivatives of f.
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Notation2 2
11 2 2
2 2
12
2 2
21
2 2
22 2 2
( )
( )
( )
( )
x x xx
x y xy
y x yx
y y yy
f f zf f f
x x x x
f f zf f f
y x y x y x
f f zf f f
x y x y x y
f f zf f f
y y y y
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Example 5Use implicit differentiation to find z/x
and z/y.
sin( , , ) 2 3x y z x y z
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Example 6 – pg. 913 # 54Find all the second partial derivatives.
2( , ) sinf x y mx ny
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Example 7Find the indicated partial derivative.
2 3( , , ) ln ; ;rss rstf r s t r rs t f f
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Clairaut’s Theorem
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Example 8 – pg. 913 # 70Find the indicated partial derivative.
6
2 3;a b c u
u x y zx y z
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More Examples
The video examples below are from section 14.3 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦Example 3◦Example 4◦Example 7
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Demonstrations
Feel free to explore these demonstrations below.
Partial Derivatives in 3DLaplace's Equation on a CircleLaplace's Equation on a Square
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