3.3 Partial Derivatives and the Gradient

Partial derivatives are key tools for understanding how multivariable functions change. They measure the rate of change with respect to one variable while keeping others constant, helping us analyze complex relationships in physics, economics, and more.

The gradient vector combines partial derivatives, pointing in the direction of steepest ascent. It's crucial for optimization problems and understanding the geometry of multivariable functions. Higher-order derivatives provide deeper insights into function behavior and curvature.

Partial Derivatives

Partial derivatives of multivariable functions

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Top images from around the web for Partial derivatives of multivariable functions

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  The Chain Rule · Calculus View original

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• Measures rate of change of function with respect to one variable while holding others constant
• Notation: $\frac{\partial f}{\partial x}$, $\frac{\partial f}{\partial y}$, $[f_x](https://www.fiveableKeyTerm:f_x)$, $[f_y](https://www.fiveableKeyTerm:f_y)$ represents partial derivatives
• Calculation steps:
  1. Treat other variables as constants
  2. Apply single-variable differentiation rules
• Common functions and their partial derivatives include polynomials ($x^2y \rightarrow 2xy$), trigonometric functions ($\sin(xy) \rightarrow y\cos(xy)$), exponential and logarithmic functions ($e^{x+y} \rightarrow e^{x+y}$)
• Chain rule extends to partial derivatives for composite functions
• Implicit differentiation applies to equations defining multivariable functions implicitly

Interpretation of partial derivatives

• Represents slope of tangent line in direction of each variable
• Relates to directional derivatives providing rates of change in specific directions
• Applications span physics (velocity components in 3D motion) and economics (marginal cost and revenue analysis)
• Enables sensitivity analysis assessing impact of small changes in variables
• Approximates small changes using linear approximation formula $\Delta f \approx f_x \Delta x + f_y \Delta y$

The Gradient and Higher-Order Derivatives

Higher-order partial derivatives

• Involve repeated differentiation with respect to same or different variables
• Notation: $\frac{\partial^2 f}{\partial x^2}$, $\frac{\partial^2 f}{\partial x \partial y}$, $f_{xy}$, $f_{yx}$ represents higher-order derivatives
• Mixed partial derivatives result from differentiating with respect to different variables
• Clairaut's theorem states equality of mixed partials under continuity conditions
• Applications include analyzing concavity (upward $f_{xx} > 0$, downward $f_{xx} < 0$) and developing Taylor series expansions for multivariable functions

Gradient vector computation and geometry

• Gradient vector $\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)$ combines all partial derivatives
• Properties: perpendicular to level curves/surfaces, points in direction of steepest ascent
• Relates to directional derivatives through dot product $\nabla f \cdot \mathbf{u}$
• Facilitates solving optimization problems (finding maxima/minima)
• Provides normal vectors to surfaces in 3D space
• Identifies conservative vector fields where curl is zero
• Components directly correspond to partial derivatives of the function