SECTION 9.3: Partial Derivatives

For a function. the influence of x and y is studied seperately on the value f (x,y) by keeping one variable fixed and varying the other. For all points (a,b):

The partial derivative of f with respect to x at (a,b) is the derivative of f with y constant.

fx (a,b) = rate of change of f with fixed y fixed at b, at the point (a,b) = lim h-0 ((f(a+(h,b)) - (f(a,b)))/ h

The partial derivative of f with respect to y at (a,b) is the derivative of f with x constant:

fy (a,b) = rate of change of f with x fixed at a, at the point (a,b) = lim h-0 ((f(a,b) + h) - (f(a,b)))/h

There is an alternative notation for Partial Derivatives:

If z = f(x,y) the we can say that,

fx (x,y) = dz/dx and fy (x,y) = dz/dy, where the d is more curvy than a regular derivative.

Partial derivatives can be used to estimate values of functions during research projects etc.

SECTION 9.4: Computing Partial Derivatives Algebraically

The second-order partial derivatives of z = f(x,y)

d^2(z)/ dx^2 = fxx = (fx)x

d^2(z)/ dxdy = fyx = (fy)x

d^2(z)/ dydx = fxy = (fx)y

d^2(z)/ dy^2 = fyy = (fy)y

It is to be noted that the mixed partial derivatives are equal, i.e., if fxy and fyx are continuous at (a,b) then fxy (a,b) = fyx (a,b)