Visualizing Partial Derivatives: f(x,y)=x^2*y at (2,3)

This section develops the concept of linear approximation and differentiability for functions of multiple variables. The differential provides a linear approximation of a function near a point, analogous to the tangent line in single-variable calculus. A function is differentiable at a point if it admits a linear approximation such that the error becomes negligible compared to the increment size. The differential is represented by the matrix of partial derivatives [f_x f_y], which maps small changes in the input to the corresponding change in the output. Unlike partial derivatives (which only measure change along axes) or directional derivatives (which only measure change along straight lines), the differential captures the function's behavior along all paths, including curved ones. This allows us to replace complex functions with their linear approximations for analysis and computation.

Differentiability measures rate of change and implies continuity. The derivative is the limit of the difference quotient. Linear approximation uses the derivative: f(x) ≈ f(a) + f'(a)(x - a). Higher-order approximations include quadratic terms. For multivariable functions, linear approximation uses partial derivatives: f(x,y) ≈ f(a,b) + (∂f/∂x)(x - a) + (∂f/∂y)(y - b). The directional derivative measures rate of change in a specific direction: D_u f = (∂f/∂x)u1 + (∂f/∂y)u2.

For functions of multiple variables, linear approximation generalizes to: L(x,y) = f(a,b) + (∂f/∂x)(a,b)(x-a) + (∂f/∂y)(a,b)(y-b). Instead of the single derivative f'(a), we use partial derivatives ∂f/∂x and ∂f/∂y evaluated at the point (a,b). Example: For f(x,y) = x·e^(xy) at (1,0), compute partial derivatives, then plug into the formula to approximate f(1.1, -0.1) ≈ 0.985, close to the exact value of 0.98542.

Linear approximation (also called local linearization or tangent line approximation) is a fundamental concept in differential calculus that approximates a function near a point using the formula f(x) ≈ f(a) + f'(a)(x-a). This concept extends naturally to multivariable functions through partial derivatives, where the linear approximation becomes f(x,y) ≈ f(a,b) + ∂f/∂x(a,b)(x-a) + ∂f/∂y(a,b)(y-b), representing the best linear approximation of the function at a given point in the plane.

This lesson covers two key concepts in multivariable calculus: (1) Linear Approximation: Using the formula f(x,y) ≈ f(a,b) + (∂f/∂x)(a,b)(x-a) + (∂f/∂y)(a,b)(y-b) to approximate function values near a point. The process involves computing partial derivatives, evaluating at the given point, and substituting into the formula. Examples include w(x,y) = x^(-3)xy + 2y^2 at (1,-1) yielding 6x - 7y - 13, and z = x²y + 3xy⁴ at (2,-1) yielding -x - 20y - 14. (2) Total Differential: Representing the change in a function when all variables change, computed as dv = (∂v/∂x)dx + (∂v/∂y)dy + (∂v/∂z)dz. Examples include v = x² - xy + (1/4)y² + 7 yielding dv = (2x - y)dx + (-x + (1/2)y)dy, and v = xy + yz + xz yielding dv = (y + z)dx + (x + z)dy + (x + y)dz.

Directional derivatives generalize partial derivatives to any direction specified by a unit vector u: D_u f(a) = lim(h→0) [f(a + hu) - f(a)]/h. The gradient vector ∇f(a) = [∂f/∂x₁(a), ..., ∂f/∂x_n(a)] contains all partial derivatives at a point. The gradient function ∇f: D → R^n assigns to each point its gradient vector. Partial derivatives are special cases of directional derivatives along coordinate axes.

The directional derivative measures the rate of change of a multivariable function in an arbitrary direction, calculated as the dot product of the gradient vector (formed by partial derivatives) and a unit direction vector. The gradient vector points in the direction of maximum increase of the function and is always perpendicular to the level curves. This concept is fundamental in physics and chemistry for understanding how quantities change in specific directions.

The directional derivative measures the rate of change of a function in an arbitrary direction, calculated as the dot product of the gradient vector and a unit vector in that direction; the gradient vector itself points in the direction of maximum rate of increase and its magnitude represents the maximum rate of change, while the opposite direction gives the maximum rate of decrease.

The directional derivative extends partial derivatives by enabling slope calculations in any direction, not just along coordinate axes. The gradient vector (∇f) is defined as a vector whose components are the partial derivatives of a function at a specific point. For f(x,y), ∇f = (∂f/∂x, ∂f/∂y); for f(x,y,z), ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z). The directional derivative is calculated as the dot product of the gradient and a unit vector in the desired direction. The direction vector must be normalized (magnitude = 1) to represent the true rate of change per unit distance. The directional derivative can also be expressed as ∂f/∂x cos(θ) + ∂f/∂y sin(θ), where θ is the angle with the x-axis. The gradient vector has three key geometric properties: (1) It is always perpendicular to level curves (curvas de nivel) of the function. (2) It always points in the direction of the greatest increase (maximum slope). (3) The magnitude of the gradient equals the maximum slope value. To find the direction of maximum increase, calculate the gradient at the point. The direction of maximum decrease is opposite to the gradient.

Directional derivatives measure the rate of change of a function at a specific point in a particular direction. The mathematical definition uses a limit: df/ds = lim(s→0) [f(x₀ + s*u₁, y₀ + s*u₂) - f(x₀, y₀)] / s. The gradient ∇f = (∂f/∂x)i + (∂f/∂y)j is a vector containing partial derivatives. The directional derivative equals ∇f · u = |∇f|cos(θ), where θ is the angle between ∇f and u. Key properties: maximum increase occurs in direction of ∇f (θ=0), maximum decrease in direction of -∇f (θ=180°), and zero change in directions orthogonal to ∇f (θ=90°). The gradient is normal to level curves, and the tangent line to level curve f(x,y) = c at (x₀, y₀) is f_x(x₀, y₀)(x - x₀) + f_y(x₀, y₀)(y - y₀) = 0. For functions of three variables f(x,y,z), the gradient is ∇f = f_x i + f_y j + f_z k, and the directional derivative in direction u = (u₁, u₂, u₃) is ∇f · u = f_x u₁ + f_y u₂ + f_z u₃.

To find local extrema of G(x, y) = 2x³ + xy² + 5x² + y², first find critical points by solving ∂G/∂x = 6x² + y² + 10x = 0 and ∂G/∂y = 2xy + 2y = 0 simultaneously. Factor ∂G/∂y = 2y(x + 1) = 0 to get y = 0 or x = -1. Substituting yields critical points at (-1, ±2), (0, 0), and (-5/3, 0). Apply the second derivative test: D = f_xx × f_yy - (f_xy)². At (0, 0): D = 20 > 0 and f_xx = 10 > 0 → local minimum. At (-1, ±2): D = -16 < 0 → saddle points. At (-5/3, 0): D = 0 → test inconclusive.

To find extrema of a multivariable function: (1) Find critical points by solving ∂f/∂x = 0 and ∂f/∂y = 0 simultaneously; (2) Compute second partial derivatives f_xx, f_yy, and f_xy; (3) Calculate D = f_xx*f_yy - (f_xy)^2 at each critical point; (4) Use the second derivative test to classify each critical point as local minimum, local maximum, or saddle point.

A critical point occurs when the gradient vector equals zero, meaning both partial derivatives with respect to x and y must be zero simultaneously. To find critical points, compute partial derivatives and set them to zero, then solve the resulting system. After finding critical points, classify them using the second derivative test: define D = f_xx * f_yy - (f_xy)². If D > 0 and f_xx > 0, it's a local minimum; if D > 0 and f_xx < 0, it's a local maximum; if D < 0, it's a saddle point (maximum in one direction, minimum in another); if D = 0, the test is inconclusive. For example, with f(x,y) = x² - 2x + y² - 4y + 5, the critical point (1,2) is a local minimum since D = 4 > 0 and f_xx = 2 > 0.

To find local extrema of multivariable functions, first locate critical points where both partial derivatives equal zero (f_x = 0 and f_y = 0), then apply the Second Derivative Test using the Hessian determinant H = f_xx * f_yy - (f_xy)^2: if H > 0 and f_xx > 0, there's a local minimum; if H > 0 and f_xx < 0, there's a local maximum; if H < 0, there's a saddle point; if H = 0, the test is inconclusive.

Compute second partial derivatives: fxx = 6x, fyy = 6y, fxy = 6. The discriminant D = fxx*fyy - (fxy)² = 36x² - 36y². Classification rules: If D > 0 and fxx > 0, relative minimum. If D > 0 and fxx < 0, relative maximum. If D < 0, saddle point. Apply to each point: (2,1) gives D = 108 > 0, fxx = 12 > 0, so relative minimum. (-2,-1) gives D = 108 > 0, fxx = -12 < 0, so relative maximum. (-1,-2) and (1,2) give D = -108 < 0, so saddle points. Substitute coordinates back into f(x,y) to find z-values. This completes the extrema analysis.