Overview
This lecture covers how to find partial derivatives for functions with multiple variables, including rules for different function types, mixed partial derivatives, and evaluating derivatives at specific points.
Basics of Partial Derivatives
- Partial derivatives measure how a function changes with respect to one variable, holding others constant.
- Denoted as ( f_x ) (partial with respect to ( x )), ( f_y ) (with respect to ( y )), etc.
- To differentiate with respect to ( x ), treat all other variables (like ( y ), ( z )) as constants.
Differentiation Rules Review
- Power Rule: ( \frac{d}{dx}x^n = nx^{n-1} ).
- Product Rule: ( (fg)' = f'g + fg' ) (use when both factors depend on the variable).
- Chain Rule (Exponentials/Logs): For ( e^{u(x)} ), ( \frac{d}{dx}e^{u(x)} = e^{u(x)}u'(x) ).
- Quotient Rule: ( (f/g)' = (gf' - fg')/g^2 ) (only if both numerator and denominator have the variable).
Examples of Partial Derivatives
- For ( f(x,y) = 7x^2 - x^3y^4 + 5x^4y^3 ):
- ( f_x = 14x - 3x^2y^4 + 20x^3y^3 )
- ( f_y = -4x^3y^3 + 15x^4y^2 )
- Exponential: For ( z = e^{x^2y^3} ):
- ( \frac{\partial z}{\partial x} = 2x y^3 e^{x^2y^3} )
- ( \frac{\partial z}{\partial y} = 3x^2 y^2 e^{x^2y^3} )
- Logarithmic: ( f(x,y) = \ln(x^2 + y^2) )
- ( f_x = \frac{2x}{x^2+y^2} )
- ( f_y = \frac{2y}{x^2+y^2} )
- Square Roots: ( f(x,y) = \sqrt{x^2 + y^2} )
- ( f_x = x/\sqrt{x^2+y^2} )
- ( f_y = y/\sqrt{x^2+y^2} )
Evaluating Partial Derivatives at Points
- To evaluate at a point, first compute the derivative, then substitute the given values.
Mixed and Higher Order Partial Derivatives
- Second partials: ( f_{xx}, f_{xy}, f_{yx}, f_{yy} ).
- Mixed partials: Order of differentiation does not matter if the function is continuous (( f_{xy} = f_{yx} )).
- Third-order and higher partials: Differentiate further, order still does not matter.
Key Terms & Definitions
- Partial Derivative — Derivative of a function with multiple variables with respect to one variable, holding others constant.
- Mixed Partial Derivative — Successive partial derivatives with respect to different variables.
- Product Rule — Formula to differentiate the product of two functions.
- Quotient Rule — Formula to differentiate a quotient of two functions.
- Chain Rule — Rule for differentiating compositions like exponentials and logs.
- Second/Third Partial Derivative — Differentiating a function twice or three times, possibly with respect to different variables.
Action Items / Next Steps
- Practice finding partial derivatives for various function types and evaluate them at given points.
- Review chain, product, and quotient rules as they apply to multivariable functions.
- Try problems involving mixed and higher order partial derivatives to reinforce understanding.