Learn Introductions to Partial Derivatives

Definition

A partial derivative measures how a multivariable function changes with respect to one variable while keeping all other variables constant. It captures the rate of change along a single dimension within a multivariable system.

What Are Partial Derivatives?

A partial derivative is written using the symbol $\partial$ instead of $d$ for regular derivatives. If a function $f(x,y)$ depends on both $x$ and $y$ , we compute:

\frac{\partial f}{\partial x} \lim_{h \rarr 0} \frac{f(x + h, y) - f(x,y)}{h} \\[6pt] \frac{\partial f}{\partial y} \lim_{h \rarr 0} \frac{f(x, y + h) - f(x,y)}{h}

Note

When differentiating with respect to one variable, treat all other variables as constants.

Computing Partial Derivatives

Consider the function:

f(x,y) = x^2y + 3y^2

Let's find, $\frac{\raisebox{1pt}{$\partial f$}}{\raisebox{-1pt}{$\partial x$}}$ :

\frac{\partial f}{\partial x} = 2xy

Differentiate with respect to $x$ , treating $y$ as a constant.

Let's compute, $\frac{\raisebox{1pt}{$\partial f$}}{\raisebox{-1pt}{$\partial y$}}$ :

\frac{\partial f}{\partial y} = x^2 + 6y

Differentiate with respect to $y$ , treating $x$ as a constant.

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Section 3. Chapter 7

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