14.3 Partial Derivatives We have stressed that a function F of two or more variables does not have a unique rate of change because each variable may affect f in different ways The partial derivatives are the rates of change with respect to each variable separately. A function f(x,y) of two variables has two partial derivatives, denoted fx and fy, defined by the following limits: fx (a; b) = lim h!0 f (a + h; b) ¡ f (a; b) f (a; b + k) ¡ f (a; b) ; fy (a; b) = lim k!0 h k Thus, fx is the derivative of f(x,b) as a function of x alone, and fy is the derivative of f(a,y) as a function of y alone. We refer to fx as the partial derivative of F with respect to X or the x-derivative Of f Partial derivatives are computed just like ordinary derivatives in one variable with this difference: to compute fx, treat y as a constant and take the derivative of f with respect to x, and to compute fy, treat x as a constant and take the derivative Of f with respect to y The differentiation rules from calculus of one variable (the product, quotient, and chain rules) are valid for partial derivatives We use the chain rule to compute partial derivatives of a composite function in the same way the chain rule is applied in a single variable case in multivariable calculus, there are a number of different ways compositions of functions can arise. Consequently, there are multiple possibilities for chain-rule derivative computations We compute the partial derivative with respect to any one of the variables by differentiating with respect to that variable while holding the remaining variables constant Higher order partial derivatives The higher order partial derivatives are the derivatives of derivatives. The second-order partial derivatives Of f are the partial derivatives of fx and fy A partial differentiation equation (PDE) is a differential equation involving Theorem 1: functions of several variables and their partial derivatives. A solution to a PDE is a function that satisfies the equation if f(xy) and f(yx) both exist and are continuous on a disk D, then f(xy) (a,b) = f(yx) (a,b) for all (a,b). Therefore on D, @2f @2f = @x@y @y@x