Partial Derivative Calculator

What is Partial Derivative Calculator?

A Partial Derivative Calculator is a specialized Derivative Calculator that performs derivative calculation for multivariable functions. While ordinary derivatives study the rate of change of a single-variable function \(y=f(x)\), partial derivatives measure how a function \(f(x,y,\dots)\) changes with respect to one variable while holding the others constant.

Formal definitions:

\[ \frac{\partial f}{\partial x}(x,y) = \lim_{h\to0}\frac{f(x+h,y)-f(x,y)}{h} \]

\[ \frac{\partial f}{\partial y}(x,y) = \lim_{k\to0}\frac{f(x,y+k)-f(x,y)}{k} \]

Higher-order and mixed partials: \[ \frac{\partial^2 f}{\partial x^2},\quad \frac{\partial^2 f}{\partial y^2},\quad \frac{\partial^2 f}{\partial x\,\partial y},\quad \frac{\partial^3 f}{\partial x^2\partial y},\ \text{etc.} \] When \(f\) is sufficiently smooth, mixed partials commute: \[ \frac{\partial^2 f}{\partial x\,\partial y} = \frac{\partial^2 f}{\partial y\,\partial x}. \]

About the Partial Derivative Calculator

This tool automates symbolic and step-by-step derivative calculation for polynomials, exponentials, logs, and trigonometric functions. It supports both ordinary derivatives (e.g., \(\frac{d}{dx}x^3\)) and partial derivatives (e.g., \(\frac{\partial}{\partial x}\sin(xy)\)). It shows each rule used—sum rule, constant rule, product/chain rules—so you can learn the process, not just the result.

Worked example (your input):

\[ \text{Find } \frac{\partial^{3}}{\partial x^{2}\partial y}\Big(e^{x}+e^{y}\Big). \] Steps:

\[ \begin{aligned} \frac{\partial}{\partial x}\!\left(e^{x}+e^{y}\right) &= \frac{\partial e^{x}}{\partial x} + \frac{\partial e^{y}}{\partial x} = e^{x} + 0 = e^{x},\\[4pt] \frac{\partial^{2}}{\partial x^{2}}\!\left(e^{x}+e^{y}\right) &= \frac{\partial}{\partial x}\!\left(e^{x}\right) = e^{x},\\[4pt] \frac{\partial^{3}}{\partial x^{2}\partial y}\!\left(e^{x}+e^{y}\right) &= \frac{\partial}{\partial y}\!\left(e^{x}\right) = 0. \end{aligned} \] Therefore, \[ \boxed{\frac{\partial^{3}}{\partial x^{2}\partial y}\!\left(e^{x}+e^{y}\right)=0.} \]

How to Use this Partial Derivative Calculator

Step 1: Enter your function \(f\) (e.g., \(f(x,y)=e^{x}+e^{y}\)).
Step 2: Choose the variable(s) to differentiate with respect to and the order (e.g., \(\partial^{3}/\partial x^{2}\partial y\)).
Step 3: Click Calculate to view a line-by-line solution using rules such as:

Sum rule: \(\ \frac{\partial}{\partial x}(u+v)=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial x}\),   Constant rule: \(\ \frac{\partial}{\partial x}(c)=0\),   Exponential: \(\ \frac{\partial}{\partial x}(e^{x})=e^{x}\),   Chain rule: \(\ \frac{\partial}{\partial x}g(h(x))=g'(h(x))h'(x)\).

The interface clearly distinguishes between ordinary derivatives and partial derivatives, ensuring accurate, transparent computations suitable for calculus, optimization, physics, and engineering.