Glossary

3.4 Chain Rule and Directional Derivatives

3 min read•july 25, 2024

The chain rule for multivariable functions extends the single-variable concept to handle complex nested functions with multiple variables. It's a powerful tool for analyzing interdependent variables, like temperature changes in heat exchangers or planetary motion in space.

Directional derivatives and gradients help us understand how functions change in specific directions. This is crucial for optimization problems, like finding the steepest path up a mountain or maximizing profit in economics. The gradient points towards the steepest increase, while directional derivatives measure change in any direction.

Chain Rule for Multivariable Functions

Chain rule for composite functions

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  • Extends single-variable chain rule to multiple variables allowing differentiation of complex nested functions
  • General form: dfdt=∂f∂xdxdt+∂f∂ydydt\frac{df}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt} computes by summing
  • Applies to functions like f(g(t),h(t))f(g(t), h(t)) where gg and hh are functions of tt
  • Crucial for analyzing interdependent variables (temperature changes in a heat exchanger)

Calculation of directional derivatives

  • Identify outer and inner functions in composite function f(g(x,y),h(x,y))f(g(x,y), h(x,y))
  • Calculate partial derivatives of outer function ∂f∂u\frac{\partial f}{\partial u} and ∂f∂v\frac{\partial f}{\partial v}
  • Compute derivatives of inner functions ∂g∂x\frac{\partial g}{\partial x}, ∂g∂y\frac{\partial g}{\partial y}, ∂h∂x\frac{\partial h}{\partial x}, ∂h∂y\frac{\partial h}{\partial y}
  • Multiply and sum results: ∂f∂x=∂f∂u∂g∂x+∂f∂v∂h∂x\frac{\partial f}{\partial x} = \frac{\partial f}{\partial u}\frac{\partial g}{\partial x} + \frac{\partial f}{\partial v}\frac{\partial h}{\partial x}
  • Repeat for ∂f∂y\frac{\partial f}{\partial y} to get complete gradient

Direction of steepest ascent/descent

  • Chain rule enables calculation of rates of change in complex systems (planetary motion)
  • Analyzes motion in multiple dimensions (projectile trajectories)
  • Optimizes multivariable functions in machine learning algorithms (gradient descent)
  • Models heat transfer in engineering applications (thermal conductivity in materials)

Gradient vs directional derivatives

  • Gradient ∇f=⟨∂f∂x,∂f∂y⟩\nabla f = \langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \rangle represents vector of partial derivatives
  • Points in direction of steepest ascent with magnitude equal to maximum rate of change
  • Duf=∇f⋅uD_\mathbf{u}f = \nabla f \cdot \mathbf{u} projects gradient onto unit vector u\mathbf{u}
  • Maximum directional derivative occurs when u\mathbf{u} aligns with ∇f\nabla f
  • Zero directional derivative when u\mathbf{u} perpendicular to ∇f\nabla f (tangent to )

Directional Derivatives and Gradients

Calculation of directional derivatives

  • Measures rate of change of function f(x,y)f(x,y) in specific direction u\mathbf{u}
  • Formula: Duf(x,y)=∇f⋅uD_\mathbf{u}f(x,y) = \nabla f \cdot \mathbf{u} combines gradient and direction vector
  • Unit vector u=⟨a,b⟩\mathbf{u} = \langle a, b \rangle where a2+b2=1a^2 + b^2 = 1 represents desired direction
  • Process:
    1. Compute gradient ∇f\nabla f
    2. Determine unit vector u\mathbf{u}
    3. Calculate dot product ∇f⋅u\nabla f \cdot \mathbf{u}
  • Interpretation:
    • Positive: function increases in given direction
    • Negative: function decreases in given direction
    • Zero: function constant in given direction (level curve)

Direction of steepest ascent/descent

  • Steepest ascent: direction of maximum increase aligns with
  • Steepest descent: direction of maximum decrease opposes gradient vector
  • Calculation:
    1. Compute gradient ∇f\nabla f at given point
    2. Normalize gradient to obtain unit vector
  • Applications:
    • Optimization in machine learning (gradient descent algorithms)
    • Path finding in computer graphics (terrain navigation)
    • Meteorology (pressure gradients in weather systems)
    • Economics (optimizing profit functions)

Gradient vs directional derivatives

  • Gradient ∇f=⟨∂f∂x,∂f∂y⟩\nabla f = \langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \rangle encapsulates all directional derivatives
  • Properties:
    • Points toward steepest ascent
    • Magnitude equals maximum rate of change
  • Directional derivative as projection: Duf=∇f⋅u=∣∇f∣cos⁡θD_\mathbf{u}f = \nabla f \cdot \mathbf{u} = |\nabla f| \cos \theta
  • Maximum directional derivative when u\mathbf{u} parallel to ∇f\nabla f
  • Zero directional derivative when u\mathbf{u} perpendicular to ∇f\nabla f
  • Gradient always perpendicular to level curves or surfaces (contour lines in topographic maps)

Key Terms to Review (15)

∇f · u: The expression ∇f · u represents the dot product of the gradient of a function and a unit vector. It measures the rate of change of the function in the direction specified by the unit vector, which is crucial for understanding how functions behave in different directions in multivariable calculus. This concept links closely to directional derivatives, showing how to compute the slope of a function as you move along a specific path defined by that unit vector.
Composition of functions: The composition of functions is an operation that takes two functions, say f and g, and combines them to create a new function, denoted as (f \circ g)(x) = f(g(x)). This means you first apply the function g to the input x and then apply the function f to the result of g. Understanding this concept is essential in multivariable calculus, as it lays the groundwork for using the chain rule and analyzing directional derivatives, which involve applying one function after another.
Continuity: Continuity is a property of functions where small changes in input lead to small changes in output. This concept is vital for ensuring that the behavior of functions remains predictable across their domain, especially when dealing with multiple dimensions and transformations. Understanding continuity helps in the analysis of limits, derivatives, and integrals, which are foundational concepts in calculus.
D_u f(x, y): The notation d_u f(x, y) represents the directional derivative of a function f at the point (x, y) in the direction of the vector u. This concept is crucial as it provides a way to measure how the function changes when moving in a specific direction, allowing for an understanding of the function's behavior beyond just its partial derivatives. The directional derivative takes into account not only the rate of change in the standard coordinate directions but also along any vector direction, making it essential for analyzing functions in multiple dimensions.
Differentiable function: A differentiable function is a function that has a derivative at every point in its domain, meaning it can be locally approximated by a linear function. This property is essential for understanding how functions change and is linked to concepts such as continuity and smoothness. In the context of multivariable calculus, differentiable functions enable the exploration of rates of change in multiple dimensions and serve as a foundation for applying techniques like the chain rule and computing directional derivatives.
Directional Derivative: The directional derivative is a measure of how a function changes as you move in a specific direction from a given point. It generalizes the concept of a derivative to multiple dimensions, allowing you to understand how functions behave when approached from various angles. This concept is deeply linked to partial derivatives and the gradient, which collectively help determine the rate of change of functions in multidimensional spaces.
Gradient method: The gradient method is a mathematical optimization technique used to find the maximum or minimum values of a function by following the direction of the steepest ascent or descent. This approach relies on calculating the gradient, which is a vector that indicates the direction and rate of fastest increase of a scalar field. By moving in the opposite direction of the gradient, one can effectively minimize a function, while moving in the direction of the gradient can be used for maximization.
Gradient vector: The gradient vector is a vector that represents the direction and rate of the steepest ascent of a scalar function. It combines all the partial derivatives of a function into a single vector, which can help in understanding how changes in multiple variables affect the function's output. This concept connects to various aspects, such as how tangent planes approximate surfaces and how directional derivatives provide insight into changing functions along specific paths.
Implicit Differentiation: Implicit differentiation is a technique used to find the derivative of a dependent variable defined implicitly by an equation involving both the dependent and independent variables. Instead of solving for one variable in terms of another, implicit differentiation allows you to differentiate both sides of an equation with respect to the independent variable, applying the chain rule when necessary. This method is especially useful when dealing with equations that cannot be easily solved for one variable.
Level Curves: Level curves are the curves on a graph that represent the set of points where a multivariable function takes on a constant value. These curves provide a visual representation of how the function behaves across its domain, allowing for an understanding of the function's gradients and directional derivatives. They are crucial in visualizing functions of two variables and analyzing changes in the function's output with respect to its inputs.
Multivariable chain rule: The multivariable chain rule is a mathematical principle used to compute the derivative of a composite function involving multiple variables. This rule allows us to differentiate functions of several variables by breaking them down into simpler components and applying derivatives in a systematic way. It connects various concepts such as partial derivatives, gradients, and the direction of change in multivariable contexts.
Partial Derivatives: Partial derivatives represent the rate of change of a multivariable function with respect to one of its variables while keeping the other variables constant. This concept is crucial for understanding how functions behave in multiple dimensions, allowing for calculations like directional derivatives and applications in vector calculus. They help describe surface properties, gradients, and integrals across various fields.
Path-dependent derivative: A path-dependent derivative refers to the concept that the rate of change of a function can vary depending on the specific path taken in the multivariable space. This characteristic highlights how, unlike single-variable calculus, where derivatives provide a unique tangent slope at each point, multivariable functions can exhibit different derivatives when approached from different directions, making it crucial for understanding behaviors in fields like optimization and physics.
Tangent Plane: A tangent plane is a flat surface that touches a curved surface at a single point, representing the local linear approximation of the curved surface at that point. This concept is fundamental in understanding how multivariable functions behave and provides insights into rates of change in multiple dimensions, connecting closely to gradients and surface representations.
Total Derivative: The total derivative of a function of multiple variables captures how the function changes as all its input variables change, not just one at a time. It represents a comprehensive way to express the sensitivity of the function to changes in its inputs and is particularly useful when applying the chain rule or computing directional derivatives, as it combines both partial derivatives and the rates of change of the input variables.
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