Partial Differentiation - created from Mind Map

Seite 1

Total differentials

Chain rule

Maximum & Minimum

Total Differentials for Two Variables for a function z = f(x, y).

Total Differentials for Three Variables for function w = f(x, y, z).

Approximation

Definition: the total differential for f is dz = df = fx(x, y)dx + fy(x, y)dy

Definition: the total differential for f is dw = df = fx(x, y, z)dx +fy(x, y, z)dy +fz(x, y, z)dz

Approximations: given small values for ∆x and ∆y, ∆z = ∆f = fx(x, y)∆x + fy(x, y)∆y, and f(x+∆x, y+∆y) ≈ f(x, y)+fx(x, y)∆x +fy(x, y)∆y.

Suppose that we have two functions f(x) and g(x) and they are both differentiable. If we define F(x) = (f∘g)(x) then the derivative of F(x) is, F′(x) = f′(g(x))g′(x) If we have y = f(u) and u = g(x) then the derivative of y is, dy/dx = (dy/du)(du/dx)

Elimination

Implicit differentiation

Lagrange multipliers

let f'(x)=0 and find critical numbers then find second derivative f''(x). Apply critical numbers in the second derivative. The function f(x) is maxima when f''(x)<0 and the function f(x) is minima when f''(x)>0. To find the maximum and minimum value we need to apply those x values in the given function

Eliminate y

solve for y and substitute

provides another way to find these extreme values