PARTIAL DIFFERENTIATION 1 PROGRAMME 14 PARTIAL DIFFERENTIATION 1

Partial differentiation Small increments

Partial differentiation Small increments

Partial differentiation First partial derivatives Second order partial derivatives

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.

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.

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.

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

Partial differentiation Small increments

Partial differentiation Small increments

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:

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