Partial Derivatives

Question 1

For what functions do we use partial differentiation?

Answer 1

f (x, y)

Question 2

How do you partially differentiate?

Answer 2

We derive with respect to each derivative at a same whilst treating the other as a constant i.e. derive for all functions of x, then derive for all functions of y and then combine.

Question 3

State the product rule for partial derivatives.

Answer 3

(df/dx)y = u (x, y) (dv/dx)y + v (x, y) (du/dx)y

(df/dy)x = u (x, y) (dv/dy)x + v (x, y) (du/dy)x

Question 4

State the quotient rule for partial derivatives.

Answer 4

(df/dx)y = (v du/dx - u dv/dx) / v2

(df/dy)x = (v du/dy - u dv/dy) / v2

Question 5

For a function of two variables, how many possible partial second derivatives are there?

Answer 5

4

Question 6

State the four partial second derivatives for f (x, y).

Answer 6

d2f/dx2
d2f/dydx
d2f/dy2
d2f/dxdy

Question 7

State Euler’s reciprocity relation and annotate.

Answer 7

d/dy(df/dx) = d/dx(df/dy)

Shows that the d2f/dydx and d2f/dxdy are equal to each other.

Question 8

What is meant by ‘total differential’?

Answer 8

df = (df/dx)dx + (df/dy)dy