Partial Derivatives

The rate of change of a function as one variable changes (the rest are fixed)

Notation

$$\underset{h\to 0}{\lim }\frac{f(x_0+h,y_0)-f(x_0,y_0)}{h}=f_x^{\prime }(x_0,y_0)=\frac{\partial f}{\partial x}(x_0,y_0)$$

Finding

• At a point
  • $f_x^{\prime }(x_0,y_0)={\frac{\text{d}}{\text{d}x}f(x,y_0)|}_{x=x_0}$
• To get a general function, instead of plunging in a specific value for the fixed variable(s), treat it like a constant
• If you find the x, and y partial derivatives, you can express the rate of change in any other direction as a linear combination of the two

Properties

• Clairaut’s Theorem
• Differentiability

Actions

• Reconstructing from Partials
• Multivar Chain Rule
• Implicit Multivar Differentiation

Approximations

• Multivar Linear Approximations
• Multivar Taylor Polynomials