Essential Concepts

The Chain Rule

• The chain rule for functions of more than one variable involves the partial derivatives with respect to all the independent variables.
• Tree diagrams are useful for deriving formulas for the chain rule for functions of more than one variable, where each independent variable also depends on other variables.

Tangent Planes and Linear Approximations

• The analog of a tangent line to a curve is a tangent plane to a surface for functions of two variables.
• Tangent planes can be used to approximate values of functions near known values.
• A function is differentiable at a point if it is ”smooth” at that point (i.e., no corners or discontinuities exist at that point).
• The total differential can be used to approximate the change in a function [latex]z=f(x_{0},y_{0})[/latex] at the point [latex](x_{0},y_{0})[/latex] for given values of [latex]\Delta{x}[/latex] and [latex]\Delta{y}[/latex].

Directional Derivatives and the Gradient

• A directional derivative represents a rate of change of a function in any given direction.
• The gradient can be used in a formula to calculate the directional derivative.
• The gradient indicates the direction of greatest change of a function of more than one variable.

Key Equations

• Chain rule, one independent variable
  [latex]\dfrac{dz}{dt}=\dfrac{\partial z}{\partial x}\cdot\dfrac{dx}{dt}+\dfrac{\partial z}{\partial y}\cdot\dfrac{dy}{dt}[/latex]
• Chain rule, two independent variables
  [latex]\begin{array}{c} \dfrac{\partial z}{\partial u} &=\dfrac{\partial z}{\partial x}\dfrac{\partial x}{\partial u}+\dfrac{\partial z}{\partial y}\dfrac{\partial y}{\partial u} \\\dfrac{\partial z}{\partial v} &=\dfrac{\partial z}{\partial x}\dfrac{\partial x}{\partial v}+\dfrac{\partial z}{\partial y}\dfrac{\partial y}{\partial v} \end{array}[/latex]
• Generalized chain rule
  [latex]\dfrac{\partial w}{\partial t_{j}}=\dfrac{\partial w}{\partial x_{1}}\dfrac{\partial x_{1}}{\partial t_{j}}+\dfrac{\partial w}{\partial x_{2}}\dfrac{\partial x_{1}}{\partial t_{j}}+\ldots+\dfrac{\partial w}{\partial x_{m}}\dfrac{\partial x_{m}}{\partial t_{j}}[/latex]
• Tangent plane
  [latex]z=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})[/latex]
• Linear approximation
  [latex]L(x,y)=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})[/latex]
• Total differential
  [latex]dz=f_{x}(x_{0},y_{0})dx+f_{y}(x_{0},y_{0})(y-y_{0})dy[/latex]
• Differentiability (two variables)
  [latex]f(x,y)=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})+E(x,y)[/latex], where the error term [latex]E[/latex] satisfies [latex]\underset{(x,y)\to (x_{0},y_{0})}{\lim}\frac{E(x,y)}{\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}}}=0[/latex]
• Differentiability (three variables)
  [latex]f(x,y,z)=f(x_{0},y_{0},z_{0})+f_{x}(x_{0},y_{0},z_{0})(x-x_{0})+f_{y}(x_{0},y_{0},z_{0})(y-y_{0})+f_{z}(x_{0},y_{0},z_{0})(z-z_{0})+E(x,y,z)[/latex], where the error term [latex]E[/latex] satisfies [latex]\underset{(x,y,z)\to (x_{0},y_{0},z_{0})}{\lim}\frac{E(x,y,z)}{\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}}}=0[/latex]
• Directional derivative (two dimensions)
  [latex]D_{\bf{u}} f(a,b)=\underset{h \to {0}}{\lim} \frac{f(a+h\cos{\theta}, b+h\sin{\theta})-f(a,b)}{h}[/latex]  or  [latex]D_{\bf{u}} f(x,y)=f_{x}(x,y)\cos{\theta}+f_{y}(x,y)\sin{\theta}[/latex]
• Gradient (two dimensions)
  [latex]\nabla f(x,y)=f_{x}(x,y){\bf{i}}+f_{y}(x,y){\bf{j}}[/latex]
• Gradient (three dimensions)
  [latex]\nabla f(x,y,z)=f_{x}(x,y,z){\bf{i}}+f_{y}(x,y,z){\bf{j}}+f_{z}(x,y,z){\bf{k}}[/latex]
• Directional derivative (three dimensions)
  [latex]\begin{array}{c} D_{\bf{u}} f(x,y,z) &=\nabla f(x,y,z)\cdot{\bf{u}} \hfill \\ \hfill &=f_{x}(x,y,z)\cos{\alpha}+f_{y}(x,y,z)\cos{\beta}+f_{z}(x,y,z)\cos{\gamma}\end{array}[/latex]

Glossary

differentiable

a function [latex]f(x,y,z)[/latex] is differentiable at [latex](x_{0},y_{0})[/latex] if [latex]f(x,y)[/latex] can be expressed in the form [latex]f(x,y)=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})+E(x,y)[/latex], where the error term [latex]E(x,y)[/latex] satisfies [latex]\underset{(x,y)\to{(x_{0},y_{0})}}{\lim}\frac{E(x,y)}{\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}}}=0[/latex]

directional derivative

the derivative of a function in the direction of a given unit vector

generalized chain rule

the chain rule extended to functions of more than one independent variable, in which each independent variable may depend on one or more other variables

gradient

the gradient of the function [latex]f(x,y)[/latex] is defined to be [latex]\nabla f(x,y)=(\partial{f}{/}\partial{x}){\bf{i}}+(\partial{f}{/}\partial{y}){\bf{j}}[/latex] which can be generalized to a function of any number of independent variables

intermediate variable

given a composition of functions (e.g., [latex]f\left(x(t),y(t)\right)[/latex]) the intermediate variables are the variables that are independent in the outer function but dependent on other variables as well; in the function [latex]f\left(x(t),y(t)\right)[/latex] the variables [latex]x[/latex] and [latex]y[/latex] are examples of intermediate variables

linear approximation

given a function [latex]f(x,y)[/latex] and a tangent plane to the function at a point [latex](x_{0},y_{0})[/latex] we can approximate [latex]f(x,y)[/latex] for points near [latex](x_{0},y_{0})[/latex] using the tangent plane formula

tangent plane

given a function [latex]f(x,y)[/latex] that is differentiable at a point [latex](x_{0},y_{0})[/latex] the equation of the tangent plane to the surface [latex]z=f(x,y)[/latex] is given by [latex]z=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})[/latex]

total differential

the total differential of the function [latex]f(x,y)[/latex] at [latex](x_{0},y_{0})[/latex] is given by the formula [latex]dz=f_{x}(x_{0},y_{0})dx+f_{y}(x_{0},y_{0})dy[/latex]

tree diagram

illustrates and derives formulas for the generalized chain rule, in which each independent variable is accounted for