1. PARTIAL DIFFERENTIATION 1
PROGRAMME 14
PARTIAL DIFFERENTIATION 1

2. Partial differentiation
Small increments

3. Partial differentiation
Small increments

4. Partial differentiation
First partial derivatives
Second order partial derivatives

5. Partial differentiation
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, keeping r constant:
This is called the first partial derivative of V with respect to h.

6. 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:
This is called the first partial derivative of V with respect to r.

7. Partial differentiation
First partial derivatives
If z(x, y) is a function of two real variables it possesses 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.

8. 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:
If the two mixed second-order derivatives are continuous then they are equal

9. Partial differentiation
Small increments

10. Partial differentiation
Small increments

11. Small increments
If V = π r2 h and r changes to r + δr and h changes to h + δh (δr and δh being small increments) then V changes to V + δV where:
and so, neglecting squares and cubes of small quantities:
That is:

12. 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