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PROGRAMME 10

PROGRAMME 10. PARTIAL DIFFERENTIATION 1. Programme 10: Partial differentiation 1. Partial differentiation Small increments. Programme 10: Partial differentiation 1. Partial differentiation Small increments. Programme 10: Partial differentiation 1. Partial differentiation

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PROGRAMME 10

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  1. PROGRAMME 10 PARTIAL DIFFERENTIATION 1

  2. Programme 10: Partial differentiation 1 Partial differentiation Small increments

  3. Programme 10: Partial differentiation 1 Partial differentiation Small increments

  4. Programme 10: Partial differentiation 1 Partial differentiation First partial derivatives Second order partial derivatives

  5. Programme 10: 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.

  6. Programme 10: 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.

  7. Programme 10: Partial differentiation 1 Partial differentiation First partial derivatives 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.

  8. Programme 10: 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

  9. Programme 10: Partial differentiation 1 Partial differentiation Small increments

  10. Programme 10: Partial differentiation 1 Small increments If V =  r2 h and r changes to r + r and h changes to h + h (r andhbeing small increments) then V changesto V + V where: and so, neglecting squares and cubes of small quantities: That is:

  11. Programme 10: Partial differentiation 1 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

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