Lecture #10: Partial Derivatives

Derivatives of Single Variable Functions

So far, everything we have discussed applies to functions of a single variable, of the form \(f(x)\). What if we want to expand this to multivariable functions?

Partial Derivatives of Functions \(f(x, y)\)

With functions \(f(x, y)\) of two variables, you have the choice of directions (a.k.a. )–which are parallel to the \(x\)-axis or the \(y\)-axis.

Example: Partial Derivatives

Examples:

1. Find \(\dfrac{\partial f}{\partial x} (2, 1)\) and \(\dfrac{\partial f}{\partial y} (2, 1)\) if \(f(x, y) = x^2 + y^2\).
2. Find \(\dfrac{\partial f}{\partial x} (2, 1)\) and \(\dfrac{\partial f}{\partial y} (2, 1)\) if \(f(x, y) = 3x + 5xy + y^2\).

Partial Derivatives of Functions \(f(x, y)\)

We can also take higher order partial derivatives: \[ \begin{aligned} \dfrac{\partial^2 f}{\partial x^2}(a, b) = f_{xx}(a, b) &= \lim_{h \to 0} \frac{\dfrac{\partial f}{\partial x}(a+h, b) - \dfrac{\partial f}{\partial x}(a, b)}{h} \\ \dfrac{\partial^2 f}{\partial y^2}(a, b) = f_{yy}(a, b) &= \lim_{h \to 0} \frac{\dfrac{\partial f}{\partial y}(a, b+h) - \dfrac{\partial f}{\partial y}(a, b)}{h} \end{aligned} \]

Partial Derivatives of Functions \(f(x, y)\)

We can also take higher order partial derivatives: \[ \begin{aligned} \dfrac{\partial^2 f}{\partial x \partial y}(a, b) = f_{xy}(a, b) &= \lim_{h \to 0} \frac{\dfrac{\partial f}{\partial x}(a, b+h) - \dfrac{\partial f}{\partial x}(a, b)}{h}\\ \dfrac{\partial^2 f}{\partial y \partial x}(a, b) = f_{yx}(a, b) &= \lim_{h \to 0} \frac{\dfrac{\partial f}{\partial y}(a+h, b) - \dfrac{\partial f}{\partial x}(a, b)}{h} \end{aligned} \]

Partial Derivatives of Functions \(f(x, y)\)

You can also extend this concept to higher dimensional functions, \(f(x_1, \dots, x_n)\): \[ \begin{aligned} \dfrac{\partial f}{\partial x_i}&(a_1, a_2, \dots, a_{n-1}, a_n) = f_{x_i}(x_1, \dots, x_n) \\ &= \lim_{h \to 0} \frac{f(a_1, \dots, a_i + h, \dots, a_n) - f(a_1, \dots, a_i, \dots, a_n)}{h} \end{aligned} \]

Partial Derivatives using Contour Plot

Example: Find \(\dfrac{\partial f}{\partial x} (2, 2)\) and \(\dfrac{\partial f}{\partial y} (2, 2)\).

Partial Derivatives using Function Table

Example: The wind chill is a function \(w(v, T)\) of the wind speed \(v\) (in \(\frac{m}{hr}\)) and temperature \(T\) (in degrees \(F\)). Estimate \(w_T(20, -15)\) and \(w_v(20, -15)\). Explain what they mean.

Partial Derivatives and Concavity

\(\dfrac{\partial f}{\partial x} = f_x(a, b)\) represents the rate of change of \(f(x, y)\) in the \(x\)-direction at the point \((a, b)\).

\(\dfrac{\partial f}{\partial y} = f_y(a, b)\) represents the rate of change of \(f(x, y)\) in the \(y\)-direction at the point \((a, b)\).

Partial Derivatives and Concavity

\(\dfrac{\partial^2 f}{\partial x^2} = f_{xx}(a, b)\) represents the concavity of \(f(x, y)\) in the \(x\)-direction at the point \((a, b)\).

\(\dfrac{\partial^2 f}{\partial y^2} = f_{yy}(a, b)\) represents the concavity of \(f(x, y)\) in the \(y\)-direction at the point \((a, b)\).

Partial Derivative Functions

If we want the function form of the partial derivative, treat the variable not being derived on as a constant and derive as if it was a single variable function.

Example: Partial Derivatives

Example: Let \(f(x, y) = yx^2 + 3xy^2\).

1. Find \(\dfrac{\partial f}{\partial x}\).
2. Find \(f_y\).

Example: Partial Derivatives

Example: Let \(g(x, y) = \dfrac{xy+y^2+x^2}{y+x}\).

1. Find \(\dfrac{\partial g}{\partial x}\).
2. Find \(g_y\).

Example: Higher-Order Partial Derivatives

Example: Let \(h(x, y) = -4 x^3 y^2\).

1. Find \(h_{xx}\).
2. Find \(h_{yy}\).
3. Find \(\dfrac{\partial^2 h}{\partial x \partial y}\).
4. Find \(\dfrac{\partial^2 h}{\partial y \partial x}\).