Partial Derivative Calculator — Theory
1. What is a partial derivative?
A partial derivative measures how a multivariable function changes with respect to one variable while the other
variables are held constant.
\[
f_x=\frac{\partial f}{\partial x},
\qquad
f_y=\frac{\partial f}{\partial y}.
\]
2. Treat the other variables as constants
If
\[
f(x,y)=x^2y+3xy^3,
\]
then when differentiating with respect to \(x\), \(y\) is treated as a constant:
\[
\begin{aligned}
\frac{\partial}{\partial x}(x^2y)&=2xy,\\
\frac{\partial}{\partial x}(3xy^3)&=3y^3.
\end{aligned}
\]
Therefore,
\[
f_x=2xy+3y^3.
\]
3. Higher-order partial derivatives
A higher-order partial derivative means differentiating more than once.
\[
f_{yy}
=
\frac{\partial^2 f}{\partial y^2}.
\]
For
\[
f(x,y)=x^2y+3xy^3,
\]
first compute
\[
f_y=x^2+9xy^2.
\]
Then differentiate again with respect to \(y\):
\[
f_{yy}=18xy.
\]
4. Mixed partial derivatives
A mixed partial derivative uses more than one variable. For example,
\[
f_{xy}
=
\frac{\partial}{\partial y}
\left(
\frac{\partial f}{\partial x}
\right).
\]
This means differentiate first with respect to \(x\), then differentiate the result with respect to \(y\).
5. Equality of mixed partials
For many smooth functions, the order of mixed partial derivatives does not matter.
\[
f_{xy}=f_{yx}.
\]
This is often called Clairaut's theorem. The function must be sufficiently smooth near the point.
6. Point evaluation
After finding a partial derivative, it can be evaluated at a point.
\[
f_x(a,b,c)
\]
This gives the instantaneous rate of change in the \(x\)-direction at that point.
7. Surface interpretation
For a two-variable function,
\[
z=f(x,y),
\]
the graph is a surface. A partial derivative describes the slope of a slice of that surface.
\[
f_x(a,b)
=
\text{slope of the slice }y=b,
\]
\[
f_y(a,b)
=
\text{slope of the slice }x=a.
\]
8. Sliced planes
The calculator's 3D graph shows two important slices:
\[
x=x_0
\qquad
\text{and}
\qquad
y=y_0.
\]
These slices make the surface easier to understand because each slice turns the surface into a one-variable curve.
9. Example calculation
Let
\[
f(x,y)=x^2y+3xy^3.
\]
For the first partial with respect to \(x\):
\[
\begin{aligned}
f_x
&=
\frac{\partial}{\partial x}(x^2y+3xy^3)\\
&=
2xy+3y^3.
\end{aligned}
\]
For the second partial with respect to \(y\):
\[
\begin{aligned}
f_y
&=
x^2+9xy^2,\\
f_{yy}
&=
18xy.
\end{aligned}
\]
11. Common mistakes
- Forgetting to hold other variables constant: when computing \(f_x\), treat \(y\) and \(z\) as constants.
- Mixing up \(f_{xy}\) and \(f_{yx}\): the order describes the order of differentiation.
- Assuming mixed partials are always equal: equality requires smoothness conditions.
- Evaluating too early: first find the symbolic derivative, then substitute the point.
- Reading a surface without slices: sliced planes make directional behavior easier to see.
