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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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14.3 Partial Derivatives 2

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.

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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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