Review of Partial Derivatives

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Page 1 Review of Partial Derivatives Definition of Partial Derivatives. Let zf x y= (,) be a function. Then the partial derivative of f with respect to x is defined as ∂ ∂ ≡≡ +− → f x fx y fx h y fx y hx h ( , ) lim (, ) ( , ) 0 and the partial derivative of f with respect to y is defined as ∂ ∂ ≡≡ +− → f y fx y fx y k fx y ky k ( , ) lim (, ) (,) 0 Partial derivatives may be computed algebraically; all of the same rules that applied to regular derivatives also apply to partial derivatives. The only trick to remember when taking a partial derivative is hold all other variables (besides the one we are differentiating with respect to) constant. Example. Let fx y x y(,) = + 2 1 . Then fx y x x yy x x x yx (,) = + = + = + ∂ ∂ ∂ ∂ 2 2 1 1 1 2 1 fx y y x y x yy x y y x yy (,) ( ) () = + = + =+ = − + −∂ ∂ ∂ ∂ ∂ ∂ 2 22 1 2 21 1 1 1 1 Example. Find the partial derivatives of fx y y e x(,) = 23 fx y x ye y x ey e y ex xx x x(,) = () = () =⋅ =∂ ∂ ∂ ∂ 23 2 3 2 3 23 33 fx y y ye e y ye y y ey xx x x(,) = () = () =⋅ =∂ ∂ ∂ ∂ 23 3 2 3 3 22 Example. Find the partial derivative of fx y z xy z(,,) = 23 fx y x xy z y zx x y z x xy zx (,) =     = () =⋅=∂ ∂ ∂ ∂ 23 3 2 33 2 2 fx y y xy z x zy y x z y xy zy (,) =     = () =⋅ =∂ ∂ ∂ ∂ 23 2 3 2 2 22 3 3 fx y z xy z xy zz xy z zx y z xy zz (,) =     =     = () =− () =−−−∂ ∂ ∂ ∂ ∂ ∂ 23 23 23 1 23 2 23 2 1 Page 2 Second Order Partial Derivatives We define higher order partial derivatives in much the same way as we did in single-variable calculus. f f x x f x f f y y f y xx yy = ∂ ∂ = ∂ ∂ ∂ ∂     = ∂ ∂ = ∂ ∂ ∂ ∂     2 2 2 2 With partial derivatives, we can also combine the variables, so there are more derivatives at each order. For example, we can differentiate fx with respect to y and we can differentiate fy with…