📖 14. Partial Derivatives Calculus

📖 From Calculus • 812 questions available

About 14. Partial Derivatives Calculus

Definition:
A partial derivative is the derivative of a function with two or more variables, taken with respect to just one variable while treating all other variables as constants. For a function $f(x, y)$, the partial derivative with respect to $x$, written as $\frac{\partial f}{\partial x}$, means you hold $y$ fixed and differentiate normally using rules like the power rule. Similarly, $\frac{\partial f}{\partial y}$ means you hold $x$ constant and differentiate with respect to $y$. For example, if $f(x, y) = 4x^3y + 2y^2$, then $\frac{\partial f}{\partial x} = 12x^2y$ (since $y$ is constant) and $\frac{\partial f}{\partial y} = 4x^3 + 4y$ (since $x^3$ is constant).

Example:
Let $f(x, y) = x^2 \sin(y) + 5y$. To find $\frac{\partial f}{\partial x}$, treat $y$ as a constant: the derivative of $x^2 \sin(y)$ is $2x \sin(y)$, and $5y$ vanishes, so $\frac{\partial f}{\partial x} = 2x \sin(y)$. To find $\frac{\partial f}{\partial y}$, treat $x$ as a constant: the derivative of $x^2 \sin(y)$ is $x^2 \cos(y)$, and the derivative of $5y$ is 5, so $\frac{\partial f}{\partial y} = x^2 \cos(y) + 5$.

Reason:
We use partial derivatives because most real-life functions depend on multiple factors—like temperature depending on both time and location, or profit depending on price and production cost. A partial derivative isolates the effect of changing exactly **one** input variable while holding everything else fixed, which answers the critical question: *“If I adjust only this one factor, how fast does the output change?”* This is essential in fields like physics for studying rates in multiple dimensions, in economics for marginal analysis of one resource, and in engineering for optimizing systems where you need to tweak one parameter at a time without interference from others."

Practice MCQs for 14. Partial Derivatives Calculus. Test your knowledge with carefully crafted questions across easy, medium, and hard difficulty levels.

🔄 Last updated: 2026-08-19