A complete set of all the lectures | MATH2019 ENGINEERING MATHEMATICS 2E | correct solutions

LECTURE 1 PARTIAL DIFFERENTIATION In your previous studies the focus was on functions of a single variable y = f(x) and their rates of change dy dx. It is however quite rare for a quantity of interest to depend on only one variable and in complicated physical systems it may be the case that the variable you are concerned with may depend upon dozens of other variables. Partial differentiation is the extension of our usual calculus to functions of several variables. Given a function of two variables z = f(x, y) we denote the rates of change in the x and y directions as ∂z ∂x and ∂z ∂y or simply as zx and zy. The formal definitions of these derivatives are presented above however in reality we only need to remember a few things to differentiate partially: The old specific rules of differentiation y y 0 x n nxn−1 e x e x sin(x) cos(x) cos(x) − sin(x) ln(x) 1 x sinh(x) cosh(x) cosh(x) sinh(x) The old general rules of differentiation (uv) 0 = u 0 v + v 0u Product Rule u v 0 = vu0 − uv0 v 2 Quotient Rule The only extra issue that needs to be kept in mind is that when you are differentiating in a particular direction you treat all other variables exactly as if they were constant. Example 1 Find ∂z ∂x and ∂z ∂y if z = x 2 + y 5 + 7. F ∂z ∂x = 2x, ∂z ∂y = 5y 4 F 2 Example 2 Suppose that z = f(x, y) = x 3 y 5 + 3x − 8y + 2. Find the function value and the rate of change of f in the x direction at the point (1, 2). F f(1, 2) = 21, ∂z ∂x(1, 2) = 99 F Example 3 Find ∂w ∂u and ∂w ∂v if w = u 3 v 4 + sinh(v 9 ) . F ∂w ∂u = 3u 2 v 4 , ∂w ∂v = 4u 3 v 3 + 9v 8 cosh(v 9 ) F