Calculus: Chain Rule and Directional Derivative

High Order Partial Derivatives

• If resistors of R1, R2, and R3 ohms are connected in parallel to make an R-ohm resistor, the value of R can be found from the equation.

Chain Rule

• The chain rule is a rule for differentiating composite functions.
• Example: If z = e^y + y tan x, where x = t + 3t and y = cosh t^2, find dz/dt.
• Example: If z = e^y + y tan x, where x = 3t + t ln s and y = 4s + cosh t^2, find ∂z/∂s and ∂z/∂t.

Implicit Derivative

• Suppose that the function F(x, y) is differentiable and the equation F(x, y) = 0 defines y implicitly as a function of x, say y = f(x).
• Since w = F(x, y) = 0, the derivative dw/dx must be zero.
• Computing the derivative from the Chain Rule, we find the derivative of the equation F(x, y) = 0.

Gradient

• The gradient of a function f(x, y) is a vector that points in the direction of the maximum rate of increase of the function.
• The gradient is defined as ∇f = (∂f/∂x)i + (∂f/∂y)j.
• The direction of the gradient is the direction of the maximum rate of increase of the function.
• The magnitude of the gradient is the maximum rate of increase of the function.

Directional Derivative

• The directional derivative is a measure of the rate of change of a function in a specific direction.
• The directional derivative can be calculated using the formula: ∂f/∂u = ∇f · u.
• The directional derivative is a dot product of the gradient and the unit vector in the direction of interest.

Function of Three Variables

• For a differentiable function f(x, y, z) and a unit vector u = u1i + u2j + u3k in space, the directional derivative can be calculated using the formula: ∂f/∂u = ∇f · u.
• Example: Find the derivative of the function f(x, y, z) = x^3 - xy^2 - z at P0 (1, 1, 0) in the direction of v = 2i - 3j + 6k.

Extrema for Functions of Two Variables

• A function f(x, y) has a local maximum value at (a, b) if f(a, b) ≥ f(x, y) for all domain points (x, y) in an open disk centered at (a, b).
• A function f(x, y) has a local minimum value at (a, b) if f(a, b) ≤ f(x, y) for all domain points (x, y) in an open disk centered at (a, b).
• A saddle point is a critical point where the function increases in one direction and decreases in another direction.

Assignment 3

• Express dw/dt as a function of t, both by using the Chain Rule.
• Find the derivative of the function at P0 in the direction of u.