1. PARTIAL DIFFERENTIATION 2
PROGRAMME 11
PARTIAL DIFFERENTIATION 2

2. Programme 11: Partial differentiation 2
Rate-of-change problems
Implicit functions
Change of variables

3. Programme 11: Partial differentiation 2
Rate-of-change problems
Implicit functions
Change of variables

4. Programme 11: Partial differentiation 2
Recall that if z(x, y) is a function of two real variables it possess two first partial derivatives.
One with respect to x,
obtained by keeping y fixed and
one with respect to y,
obtained by keeping x fixed.
All the usual rules for differentiating sums, differences, products, quotients and functions of a function apply.

5. Programme 11: Partial differentiation 2
Recall further that if both x and y vary simultaneously then, for small changes x and y:

6. Programme 11: Partial differentiation 2
Rate-of-change problems
Implicit functions
Change of variables

7. Programme 11: Partial differentiation 2
Rate-of-change problems
If the variations in x and y occur during the time interval t then:
So that, in the limit as t  0

8. Programme 11: Partial differentiation 2
Rate-of-change problems
Implicit functions
Change of variables

9. Programme 11: Partial differentiation 2
Implicit functions
Partial differentiation can be used to find the derivatives of implicit functions. If f (x, y) = 0 is an expression involving x and y where y is assumed to be an implicit function of x then letting z = f (x, y):
In the limit as x  0:
Therefore:

10. Programme 11: Partial differentiation 2
Rate-of-change problems
Implicit functions
Change of variables

11. Programme 11: Partial differentiation 2
Change of variables
If z = z(x, y) where x = x(u, v) and y = y(u, v) then:
and as u  0 so:
Similarly:

12. Programme 11: Partial differentiation 2
Learning outcomes
Derive the first- and second order partial derivatives of a function of two real variables
Apply partial differentiation to rate-of-change problems
Apply partial differentiation to implicit functions
Apply partial differentiation to change-of-variable problems