Programme 14 Partial differentiation 1 PROGRAMME 14 PARTIAL

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Programme 14: Partial differentiation 1 PROGRAMME 14 PARTIAL DIFFERENTIATION 1 STROUD Worked examples and

Programme 14: Partial differentiation 1 PROGRAMME 14 PARTIAL DIFFERENTIATION 1 STROUD Worked examples and exercises are in the text

Programme 14: Partial differentiation 1 Partial differentiation Small increments STROUD Worked examples and exercises

Programme 14: Partial differentiation 1 Partial differentiation Small increments STROUD Worked examples and exercises are in the text

Programme 14: Partial differentiation 1 Partial differentiation Small increments STROUD Worked examples and exercises

Programme 14: Partial differentiation 1 Partial differentiation Small increments STROUD Worked examples and exercises are in the text

Programme 14: Partial differentiation 1 Partial differentiation First partial derivatives Second order partial derivatives

Programme 14: Partial differentiation 1 Partial differentiation First partial derivatives Second order partial derivatives STROUD Worked examples and exercises are in the text

Programme 14: Partial differentiation 1 Partial differentiation First partial derivatives The volume V of

Programme 14: Partial differentiation 1 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. STROUD Worked examples and exercises are in the text

Programme 14: Partial differentiation 1 Partial differentiation First partial derivatives Similarly, if h is

Programme 14: Partial differentiation 1 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. STROUD Worked examples and exercises are in the text

Programme 14: Partial differentiation 1 Partial differentiation First partial derivatives If z(x, y) is

Programme 14: Partial differentiation 1 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. STROUD Worked examples and exercises are in the text

Programme 14: Partial differentiation 1 Partial differentiation Second-order partial derivatives The first partial derivatives

Programme 14: Partial differentiation 1 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 STROUD Worked examples and exercises are in the text

Programme 14: Partial differentiation 1 Partial differentiation Small increments STROUD Worked examples and exercises

Programme 14: Partial differentiation 1 Partial differentiation Small increments STROUD Worked examples and exercises are in the text

Programme 14: Partial differentiation 1 Partial differentiation Small increments STROUD Worked examples and exercises

Programme 14: Partial differentiation 1 Partial differentiation Small increments STROUD Worked examples and exercises are in the text

Programme 14: Partial differentiation 1 Small increments If V = π r 2 h

Programme 14: Partial differentiation 1 Small increments If V = π r 2 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: STROUD Worked examples and exercises are in the text

Programme 14: Partial differentiation 1 Learning outcomes üFind the first partial derivatives of a

Programme 14: Partial differentiation 1 Learning outcomes üFind the first partial derivatives of a function of two real variables üFind the second-order partial derivatives of a function of two real variables üCalculate errors using partial differentiation STROUD Worked examples and exercises are in the text