Prog 10 Partial 1 MEF

PARTIAL
DIFFERENTIATION

STROUD

Worked examples and exercises are in the text

Partial differentiation
Partial differentiation

Small increments

STROUD

Worked examples and exercises are in the text

Partial differentiation
Partial differentiation

Small increments

STROUD

Worked examples and exercises are in the text

Partial differentiation
Partial differentiation
First partial derivatives
Second order partial derivatives

STROUD

Worked examples and exercises are in the text

Partial differentiation
Partial differentiation
First partial derivatives

The volume V of a cylinder of radius r and height
h is given by:
V   r 2h

If r is kept constant and h increases then V
increases. We can find the rate of change of V with
respect to h by differentiating with respect to h,
keeping r constant:
 dV 
2

r

we write this as
 dh 
r constant

V
r2
h

This is called the first partial derivative of V with
respect to h.

STROUD

Worked examples and exercises are in the text

Partial differentiation
Partial differentiation
First partial derivatives

Similarly, if h is kept constant and r increases then again, V increases. We
can then find the rate of change of V by differentiating with respect to r
keeping h constant:

 dV 
 2 rh we write this as
 dr 
h constant

V
 2 rh

r

This is called the first partial derivative of V with respect to r.

STROUD

Worked examples and exercises are in the text

Partial differentiation
Partial differentiation
First partial derivatives

If z(x, y) is a function of two real variables it possess two first partial
derivatives.
One with respect to x,

one with respect to y,

z
x

z
y

obtained by keeping y fixed and

obtained by keeping x fixed.

All the usual rules for differentiating sums, differences, products,
quotients and functions of a function apply.

STROUD

Worked examples and exercises are in the text

Partial differentiation
Partial differentiation
Second-order partial derivatives

The first partial derivatives of a function of two variables are each

themselves likely to be functions of two variables and so can themselves
be differentiated. This gives rise to four second-order partial derivatives:

  z   2 z
 
x  x  x2

  z   2 z
  2
y  y  y

  z   2 z
 
y  x  yx

  z   2 z
 
x  y  xy

If the two mixed second-order derivatives are continuous then they are

equal
2 z
2 z

yx xy

STROUD

Worked examples and exercises are in the text

Partial differentiation
Partial differentiation

Small increments

STROUD

Worked examples and exercises are in the text

Partial differentiation

Small increments

If V =  r 2 h and r changes to r + r and h changes to h + h (r and h
being small increments) then V changes to V + V where:
V   V   (r   r ) 2 (h   h)

  r 2 h   r 2 h  2 r  rh  2 r  r  h   r  h   r   h
2

2

 V   r 2 h  2 rh r  2 r  h r   r  h   r   h
2

2

and so, neglecting squares and cubes of small quantities:
That is:

STROUD

 V  2 rh r   r 2 h

V 

V
V
r 
h
r
h

Worked examples and exercises are in the text

Partial differentiation
Learning outcomes

Find the first partial derivatives of a function of two real variables
Find the second-order partial derivatives of a function of two real variables
Calculate errors using partial differentiation

STROUD

Worked examples and exercises are in the text