Understanding Partial Derivatives: What They Are + Simple Problems

In single-variable calculus, we learn how to find the rate of change of a function with respect to one variable. But what happens when a function depends on multiple variables? That's where partial derivatives come in!

Partial derivatives allow us to examine how a multivariable function changes when only one of its independent variables changes, while keeping the others constant.

Why Do We Need Partial Derivatives?

Imagine a function that describes the temperature on a metal plate, $T(x, y)$, where $x$ and $y$ are spatial coordinates. If you want to know how the temperature changes as you move only along the x-axis (without changing your y-position), you'd use a partial derivative. Similarly, you could find the rate of change along the y-axis.

Definition and Notation

For a function $f(x, y)$ of two variables, the partial derivative with respect to $x$ is denoted by:

$$\frac{\partial f}{\partial x} \quad \text{or} \quad f_x(x, y)$$

And the partial derivative with respect to $y$ is denoted by:

$$\frac{\partial f}{\partial y} \quad \text{or} \quad f_y(x, y)$$

The symbol $\partial$ (pronounced "del" or "partial") distinguishes a partial derivative from an ordinary derivative.

How to Compute Partial Derivatives

The trick to computing partial derivatives is to treat all other variables as if they were constants.

To find $\frac{\partial f}{\partial x}$:

1. Treat $y$ (and any other variables) as constants.
2. Differentiate $f(x, y)$ with respect to $x$ using all the usual differentiation rules (power rule, product rule, chain rule, etc.).

To find $\frac{\partial f}{\partial y}$:

1. Treat $x$ (and any other variables) as constants.
2. Differentiate $f(x, y)$ with respect to $y$.

Simple Problem 1: Basic Polynomial Function

Let $f(x, y) = 3x^2y + 2xy^3 + 5x - 7y + 10$.

Find $\frac{\partial f}{\partial x}$: Treat $y$ as a constant.

• The derivative of $3x^2y$ with respect to $x$ is $3y \cdot (2x) = 6xy$.
• The derivative of $2xy^3$ with respect to $x$ is $2y^3 \cdot (1) = 2y^3$.
• The derivative of $5x$ with respect to $x$ is $5$.
• The derivative of $-7y$ with respect to $x$ is $0$ (since $-7y$ is treated as a constant).
• The derivative of $10$ with respect to $x$ is $0$.

So,

$$\frac{\partial f}{\partial x} = 6xy + 2y^3 + 5$$

Find $\frac{\partial f}{\partial y}$: Treat $x$ as a constant.

• The derivative of $3x^2y$ with respect to $y$ is $3x^2 \cdot (1) = 3x^2$.
• The derivative of $2xy^3$ with respect to $y$ is $2x \cdot (3y^2) = 6xy^2$.
• The derivative of $5x$ with respect to $y$ is $0$.
• The derivative of $-7y$ with respect to $y$ is $-7$.
• The derivative of $10$ with respect to $y$ is $0$.

So,

$$\frac{\partial f}{\partial y} = 3x^2 + 6xy^2 - 7$$

Simple Problem 2: Exponential and Trigonometric Function

Let $g(x, y) = e^{xy} + \cos(x^2y)$.

Find $\frac{\partial g}{\partial x}$: Treat $y$ as a constant.

• For $e^{xy}$: Use the chain rule. The derivative of $e^u$ is $e^u \cdot u'$. Here $u = xy$, so $u' = y$ (derivative with respect to $x$). So, the derivative is $y e^{xy}$.
• For $\cos(x^2y)$: Use the chain rule. The derivative of $\cos(u)$ is $-\sin(u) \cdot u'$. Here $u = x^2y$, so $u' = 2xy$ (derivative with respect to $x$). So, the derivative is $-\sin(x^2y) \cdot 2xy$.

Thus,

$$\frac{\partial g}{\partial x} = y e^{xy} - 2xy \sin(x^2y)$$

Find $\frac{\partial g}{\partial y}$: Treat $x$ as a constant.

• For $e^{xy}$: Use the chain rule. The derivative of $e^u$ is $e^u \cdot u'$. Here $u = xy$, so $u' = x$ (derivative with respect to $y$). So, the derivative is $x e^{xy}$.
• For $\cos(x^2y)$: Use the chain rule. The derivative of $\cos(u)$ is $-\sin(u) \cdot u'$. Here $u = x^2y$, so $u' = x^2$ (derivative with respect to $y$). So, the derivative is $-\sin(x^2y) \cdot x^2$.

Thus,

$$\frac{\partial g}{\partial y} = x e^{xy} - x^2 \sin(x^2y)$$

Higher-Order Partial Derivatives (Briefly)

Just like ordinary derivatives, you can take partial derivatives multiple times. For example, $\frac{\partial^2 f}{\partial x^2}$ means you take the partial derivative with respect to $x$ twice. You can also take "mixed" partial derivatives like $\frac{\partial^2 f}{\partial y \partial x}$, which means you first differentiate with respect to $x$, and then with respect to $y$.

Real-World Applications

Partial derivatives are crucial in many fields:

• Physics: Thermodynamics, fluid dynamics, electromagnetism.
• Engineering: Structural analysis, control systems.
• Economics: Marginal utility, elasticity.
• Machine Learning: Gradient descent algorithms to optimize models.

Key Takeaways

• Partial derivatives measure the rate of change of a multivariable function with respect to one variable, holding others constant.
• To compute them, treat all other variables as constants.
• They are denoted by $\partial$ (e.g., $\frac{\partial f}{\partial x}$).
• Essential for understanding how complex systems change.

Keep practicing, and you'll master them in no time!