MULTIVARIABLE CALCULUS 02
Partial Derivatives
A partial derivative measures how a multivariable function changes when one input changes while the other inputs are held fixed.
The main idea
Suppose
\[ z=f(x,y). \]The output \(z\) depends on both \(x\) and \(y\).
We can ask two different questions:
Partial derivatives answer these questions.
1. Partial derivative with respect to \(x\)
The partial derivative of \(f\) with respect to \(x\) is written
or
\[ f_x. \]When differentiating with respect to \(x\), treat \(y\) as a constant.
2. Partial derivative with respect to \(y\)
Similarly,
or
\[ f_y \]measures change with respect to \(y\) while \(x\) is held fixed.
3. A simple example
Let
\[ f(x,y)=x^2+3xy+y^2. \]To find
\[ \frac{\partial f}{\partial x}, \]treat \(y\) as constant.
Differentiate term by term:
\[ \frac{\partial}{\partial x}(x^2)=2x, \] \[ \frac{\partial}{\partial x}(3xy)=3y, \]and
\[ \frac{\partial}{\partial x}(y^2)=0. \]Therefore,
4. Now differentiate with respect to \(y\)
This time treat \(x\) as constant.
\[ \frac{\partial}{\partial y}(x^2)=0, \] \[ \frac{\partial}{\partial y}(3xy)=3x, \]and
\[ \frac{\partial}{\partial y}(y^2)=2y. \]Therefore,
5. Why the symbol changes from \(d\) to \(\partial\)
For a function of one variable,
\[ y=f(x), \]we use
\[ \frac{dy}{dx}. \]For a function of several variables,
\[ z=f(x,y), \]we use the partial-derivative symbol
\[ \partial. \]It reminds us that only one source of variation is being considered at a time.
6. Geometric meaning
Suppose
\[ z=f(x,y) \]defines a surface.
If we fix \(y=y_0\), we obtain a curve through the surface:
\[ z=f(x,y_0). \]Then
\[ \frac{\partial f}{\partial x}(x_0,y_0) \]is the slope of that curve at \((x_0,y_0)\).
7. Fixing one variable creates an ordinary function
Consider
\[ f(x,y)=x^2+y^2. \]If we fix
\[ y=2, \]then
\[ f(x,2)=x^2+4. \]This is now an ordinary one-variable function of \(x\).
Its derivative is
\[ 2x. \]Therefore,
\[ \frac{\partial f}{\partial x}=2x. \]8. Surface-slice diagram
9. Evaluate at a point
Suppose
\[ f(x,y)=x^2+3xy+y^2. \]We found
\[ f_x=2x+3y. \]At
\[ (x,y)=(1,2), \]we have
\[ f_x(1,2) = 2(1)+3(2) = 8. \]So the instantaneous rate of change in the \(x\)-direction at that point is
\[ \boxed{8}. \]10. The rate depends on direction
At the same point,
\[ f_y=3x+2y. \]Therefore,
\[ f_y(1,2) = 3+4 = 7. \]So at \((1,2)\),
\[ f_x=8, \qquad f_y=7. \]The function changes at different rates in the two coordinate directions.
11. Partial derivatives from first principles
The partial derivative with respect to \(x\) can be defined by a limit:
Notice that \(y\) remains fixed.
Similarly,
\[ \boxed{ \frac{\partial f}{\partial y}(x,y) = \lim_{h\to0} \frac{ f(x,y+h)-f(x,y) }{h} } \]12. Same derivative idea, different direction
Ordinary derivatives measure change along a one-dimensional input line.
Partial derivatives apply the same limiting idea along selected coordinate directions.
13. Second partial derivatives
A partial derivative can itself be differentiated again.
For example,
\[ f_{xx} = \frac{\partial^2 f}{\partial x^2} \]means differentiate twice with respect to \(x\).
Similarly,
\[ f_{yy} = \frac{\partial^2 f}{\partial y^2}. \]14. Mixed partial derivatives
We can also differentiate first with respect to one variable and then the other.
For example,
\[ f_{xy} = \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right). \]And
\[ f_{yx} = \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial y} \right). \]15. Example of mixed partials
Let
\[ f(x,y)=x^2y+xy^2. \]First,
\[ f_x=2xy+y^2. \]Differentiate with respect to \(y\):
\[ f_{xy}=2x+2y. \]Now begin with
\[ f_y=x^2+2xy. \]Differentiate with respect to \(x\):
\[ f_{yx}=2x+2y. \]So
\[ f_{xy}=f_{yx}. \]16. When mixed partials are equal
For sufficiently smooth functions, the mixed partial derivatives are equal:
This is an important result, but it requires suitable regularity conditions.
17. Partial derivatives in modelling
Suppose a quantity
\[ R(\beta,\gamma) \]depends on two parameters.
Then
\[ \frac{\partial R}{\partial\beta} \]measures the sensitivity of \(R\) to changes in \(\beta\) while \(\gamma\) is fixed.
Similarly,
\[ \frac{\partial R}{\partial\gamma} \]measures sensitivity to \(\gamma\) while \(\beta\) is fixed.
This makes partial derivatives fundamental in parameter sensitivity analysis.
18. The important distinction
If \(z=f(x,y)\), then
\[ \frac{\partial z}{\partial x} \]does not describe every possible way \(z\) can change.
It describes only the change produced by varying \(x\) while holding \(y\) fixed.
If both \(x\) and \(y\) change together, we need a broader idea.
19. This leads to the total differential
For a small change in both inputs,
\[ dx \qquad\text{and}\qquad dy, \]the corresponding linear change in \(z\) is
This combines the separate effects of changes in both variables.
20. The central idea
Partial derivatives extend ordinary differentiation to functions with several independent variables.