rules of partial differentiation
to find fx, regard y as a constant and differential f(x,y) with respect to x
Partial differentiation is a process of finding the derivative of a function with respect to one of its variables while keeping all other variables constant. The rules of partial differentiation are as follows:
1. Power rule: If f(x, y) = x^n where n is a constant, then ∂f/∂x = nx^(n-1) and ∂f/∂y = 0.
2. Product rule: If f(x, y) = u(x,v)y where u(x,v) and v(y) are functions of x and y respectively, then ∂f/∂x = u_x*y + u*v_x and ∂f/∂y = u*v_y.
3. Quotient rule: If f(x,y) = u(x,v) / v(y) where u(x,v) and v(y) are functions of x and y, respectively, then
∂f/∂x = [v(y)*u_x-y*u*v_x] / [v(y)]^2
and
∂f/∂y = [-u*v_y] / [v(y)]^2
4. Chain rule: If f(x,y) = g(u,v), and u=u(x,y) and v=v(x,y), then set z=u(x,y) and w=v(x,y), so that f=g(z,w). Then,
∂f/∂x = ∂g/∂z * ∂z/∂x + ∂g/∂w * ∂w/∂x
and
∂f/∂y = ∂g/∂z * ∂z/∂y + ∂g/∂w * ∂w/∂y
5. Sum rule: If f(x,y)=g(x,y)+h(x,y), then ∂f/∂x = ∂g/∂x + ∂h/∂x and ∂f/∂y = ∂g/∂y + ∂h/∂y
These rules are essential for solving problems in fields such as calculus, physics, and engineering that require differentiating functions with respect to multiple variables.
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