Partial derivatives

Definition and Notation

For a function $f(x, y, z, …)$ of multiple variables, the partial derivative with respect to a single variable measures the rate of change of the function with respect to that variable while holding all other variables constant.

Notation for Partial Derivatives

For a function $f(x, y)$:

• $\frac{\partial f}{\partial x}$ or $f_x$ denotes the partial derivative with respect to $x$
• $\frac{\partial f}{\partial y}$ or $f_y$ denotes the partial derivative with respect to $y$

Higher-Order Partial Derivatives

• Second-order partial derivatives:
  • $\frac{\partial^2 f}{\partial x^2}$ or $f_{xx}$ (second derivative with respect to $x$)
  • $\frac{\partial^2 f}{\partial y^2}$ or $f_{yy}$ (second derivative with respect to $y$)
  • $\frac{\partial^2 f}{\partial x \partial y}$ or $f_{xy}$ (derivative first with respect to $y$, then with respect to $x$)
  • $\frac{\partial^2 f}{\partial y \partial x}$ or $f_{yx}$ (derivative first with respect to $x$, then with respect to $y$)
• For “nice” functions (continuous second derivatives), mixed partial derivatives are equal: $f_{xy} = f_{yx}$

Computing Partial Derivatives

To compute the partial derivative of $f$ with respect to $x$:

1. Treat all variables other than $x$ as constants
2. Differentiate $f$ with respect to $x$ using the standard rules of differentiation

To compute the partial derivative of $f$ with respect to $y$:

1. Treat all variables other than $y$ as constants
2. Differentiate $f$ with respect to $y$ using the standard rules of differentiation

Basic Rules of Partial Differentiation

Let $u(x,y)$ and $v(x,y)$ be functions with continuous partial derivatives, and let $c$ be a constant.

1. Constant Rule: $\frac{\partial c}{\partial x} = 0$

2. Sum Rule: $\frac{\partial}{\partial x}[u(x,y) + v(x,y)] = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial x}$

3. Difference Rule: $\frac{\partial}{\partial x}[u(x,y) - v(x,y)] = \frac{\partial u}{\partial x} - \frac{\partial v}{\partial x}$

4. Product Rule: $\frac{\partial}{\partial x}[u(x,y) \cdot v(x,y)] = \frac{\partial u}{\partial x} \cdot v(x,y) + u(x,y) \cdot \frac{\partial v}{\partial x}$

5. Quotient Rule: $\frac{\partial}{\partial x}\left[\frac{u(x,y)}{v(x,y)}\right] = \frac{\frac{\partial u}{\partial x} \cdot v(x,y) - u(x,y) \cdot \frac{\partial v}{\partial x}}{[v(x,y)]^2}$

6. Chain Rule for Partial Differentiation: If $z = f(u,v)$ where $u = g(x,y)$ and $v = h(x,y)$, then: $\frac{\partial z}{\partial x} = \frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial x} + \frac{\partial f}{\partial v} \cdot \frac{\partial v}{\partial x}$

Partial Derivatives of Common Functions

Polynomials and Algebraic Expressions

For $f(x,y) = x^ny^m$:

• $\frac{\partial f}{\partial x} = nx^{n-1}y^m$
• $\frac{\partial f}{\partial y} = mx^ny^{m-1}$

Exponential Functions

For $f(x,y) = e^{ax+by}$:

• $\frac{\partial f}{\partial x} = ae^{ax+by}$
• $\frac{\partial f}{\partial y} = be^{ax+by}$

Logarithmic Functions

For $f(x,y) = \ln(x^2+y^2)$:

• $\frac{\partial f}{\partial x} = \frac{2x}{x^2+y^2}$
• $\frac{\partial f}{\partial y} = \frac{2y}{x^2+y^2}$

Trigonometric Functions

For $f(x,y) = \sin(x+y)$:

• $\frac{\partial f}{\partial x} = \cos(x+y)$
• $\frac{\partial f}{\partial y} = \cos(x+y)$

For $f(x,y) = \cos(xy)$:

• $\frac{\partial f}{\partial x} = -\sin(xy) \cdot y$
• $\frac{\partial f}{\partial y} = -\sin(xy) \cdot x$

Applications of Partial Derivatives

Tangent Planes

The equation of the tangent plane to the surface $z = f(x,y)$ at the point $(x_0, y_0, f(x_0,y_0))$ is:

Directional Derivatives

The directional derivative of $f(x,y)$ at the point $(x_0,y_0)$ in the direction of the unit vector $\mathbf{u} = (a,b)$ is:

Gradient

The gradient of $f(x,y)$ is the vector:

The gradient points in the direction of the maximum rate of increase of the function.

Critical Points

Critical points of $f(x,y)$ occur where $\nabla f(x,y) = \mathbf{0}$, i.e., where:

Worked Examples

Example 1: Basic Computation

Find the partial derivatives of $f(x,y) = x^2y^3 + e^{xy}$.

Solution:

• $\frac{\partial f}{\partial x} = 2xy^3 + ye^{xy}$
• $\frac{\partial f}{\partial y} = 3x^2y^2 + xe^{xy}$

Example 2: Ln Function

Find the partial derivatives of $f(x,y) = \ln(x^4y^3) + e^{2xy}$.

Solution: Using properties of logarithms: $\ln(x^4y^3) = \ln(x^4) + \ln(y^3) = 4\ln(x) + 3\ln(y)$

• $\frac{\partial f}{\partial x} = \frac{4}{x} + 2ye^{2xy}$
• $\frac{\partial f}{\partial y} = \frac{3}{y} + 2xe^{2xy}$

Example 3: Gradient Calculation

Determine the gradient of $f(x,y) = e^x(2x^2-xy+y^2)$ at the point (0,0).

Solution: First, we find the partial derivatives:

$\frac{\partial f}{\partial x} = e^x(2x^2-xy+y^2) + e^x(4x-y)$ $= e^x[(2x^2-xy+y^2) + (4x-y)]$ $= e^x(2x^2-xy+y^2+4x-y)$

$\frac{\partial f}{\partial y} = e^x(-x+2y)$

At the point (0,0): $\frac{\partial f}{\partial x}(0,0) = e^0(0+0+0+0-0) = 1 \cdot 0 = 0$ $\frac{\partial f}{\partial y}(0,0) = e^0(-0+0) = 1 \cdot 0 = 0$

Therefore, $\nabla f(0,0) = (0,0)$, which means (0,0) is a critical point.

Example 4: Composite Functions

Find the partial derivatives of $f(x,y) = \cos(x^2y)$.

Solution:

• $\frac{\partial f}{\partial x} = -\sin(x^2y) \cdot 2xy$
• $\frac{\partial f}{\partial y} = -\sin(x^2y) \cdot x^2$

Example 5: Mixed Second Derivatives

Find the mixed second partial derivatives of $f(x,y) = x^3y + y^2$.

Solution: First, find the first partial derivatives:

• $\frac{\partial f}{\partial x} = 3x^2y$
• $\frac{\partial f}{\partial y} = x^3 + 2y$

Now find the mixed second derivatives:

• $\frac{\partial^2 f}{\partial y \partial x} = 3x^2$
• $\frac{\partial^2 f}{\partial x \partial y} = 3x^2$

As expected, $f_{xy} = f_{yx}$.

Strategy for Exercises

1. Domain analysis: Begin by determining the domain of the function to identify any restrictions.

2. Methodical differentiation:
   • Treat all variables except the one you’re differentiating with respect to as constants
   • Apply standard rules of differentiation
   • Simplify your results
3. For complex expressions:
   • Break down the function using sum, product, or chain rules
   • Differentiate component by component
   • Combine results carefully
4. For logarithmic expressions like $\ln(x^4y^3)$:
   • Use logarithm properties to simplify: $\ln(x^4y^3) = 4\ln(x) + 3\ln(y)$
   • Then differentiate term by term
5. For exponential expressions like $e^{2xy}$:
   • Treat the exponent as a composite function
   • Apply the chain rule
   • Remember that $\frac{d}{dx}e^u = e^u \cdot \frac{du}{dx}$
6. For trigonometric functions like $\cos(x^2y)$:
   • Apply the chain rule
   • Differentiate the inner function appropriately
7. Verification:
   • Check dimensions and units
   • Test with simple values when possible
   • For mixed derivatives, verify that $f_{xy} = f_{yx}$ as a sanity check

See:

• Gradients
• Directional Derivatives
• Critical Points (Multivariable)
• Local Extrema
• Multivariable Calculus

Exercises

1. Compute the partial derivatives of the function $f(x,y) = \ln(x^4y^3) + e^{2xy}$.

2. Determine the gradient of $f(x,y) = e^x(2x^2-xy+y^2)$.

3. Determine the domain and compute the partial derivatives of the following functions

   • a) $f(x, y) = x^2+y^2$
   • b) $f(x, y) = x^3y+y^2$
   • c) $f(x, y) = \cos(x^2y)$
   • d) $f(x, y) = e^{x^2y}$
   • e) $f(x, y) = \ln(x^2+y^3) + \sqrt{xy}$
   • f) $f(x, y) = \ln(1-x-y)$
   • g) $f(x, y) = y^2+3xy+x+y-2$
   • h) $f(x, y) = x\sin(xy)$
   • i) $f(x, y) = \frac{2y}{y+\cos x}$
   • j) $f(x, y) = e^{xy}+\ln(x+y)$