Prog 10 Partial 1 MEF

STROUD

Worked examples and exercises are in the text

First partial derivatives

First partial derivatives

The volume V of a cylinder of radius r and height

h is given by:

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,

STROUD

Worked examples and exercises are in the text

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

First partial derivatives

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

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:

STROUD

Worked examples and exercises are in the text

If V =  r2 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:

and so, neglecting squares and cubes of small quantities:

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