Training Multivariable Calculus Partial Derivatives
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Partial Derivatives

26 min Multivariable Calculus

Partial Derivatives

Partial Derivative

For $f(x,y)$, the partial derivative with respect to $x$ treats $y$ as a constant:

$$f_x = \frac{\partial f}{\partial x} = \lim_{h\to 0} \frac{f(x+h,y)-f(x,y)}{h}$$

Similarly $f_y = \frac{\partial f}{\partial y}$ treats $x$ as constant.

Higher-Order Partials

$$f_{xx} = \frac{\partial^2 f}{\partial x^2}, \quad f_{yy} = \frac{\partial^2 f}{\partial y^2}, \quad f_{xy} = \frac{\partial^2 f}{\partial y \partial x}$$

Clairaut's Theorem: If $f_{xy}$ and $f_{yx}$ are continuous, then $f_{xy} = f_{yx}$.

Example 1

Find $f_x$ and $f_y$ for $f(x,y) = x^3 + 2xy - y^2$.

$f_x = 3x^2 + 2y$

$f_y = 2x - 2y$

Example 2

Find $f_x$ for $f(x,y) = e^{xy}$.

$f_x = ye^{xy}$

Example 3

Find $f_{xx}$ and $f_{xy}$ for $f(x,y) = x^2y + 3xy^3$.

$f_x = 2xy + 3y^3$. Then $f_{xx} = 2y$ and $f_{xy} = 2x + 9y^2$.

Practice Problems

1. Find $f_x$ and $f_y$: $f = x^2 + 3xy + y^3$.
2. Find $f_x$: $f = \sin(xy)$.
3. Find $f_y$: $f = \ln(x^2+y)$.
4. Find $f_{xx}$: $f = x^4 - 2x^2y$.
5. Verify $f_{xy} = f_{yx}$: $f = x^2y^3$.
6. Find $f_x$: $f = \frac{x}{x+y}$.
7. Evaluate $f_x(1,2)$: $f = x^2y - 3y^2$.
8. Find all first partials: $f(x,y,z) = xyz$.
9. Find $f_y$: $f = e^{x+2y}$.
10. Find $f_{yy}$: $f = x^2\cos y$.
11. Chain rule: $z = x^2+y^2$, $x = t$, $y = t^2$. Find $dz/dt$.
12. Find the second-order partials of $f = xy + e^x$.
Show Answer Key

1. $f_x = 2x+3y$, $f_y = 3x+3y^2$

2. $y\cos(xy)$

3. $\frac{1}{x^2+y}$

4. $f_x = 4x^3-4xy$; $f_{xx} = 12x^2-4y$

5. $f_{xy} = 6xy^2 = f_{yx}$ ✓

6. $\frac{y}{(x+y)^2}$

7. $f_x = 2xy$; $f_x(1,2) = 4$

8. $f_x = yz$, $f_y = xz$, $f_z = xy$

9. $2e^{x+2y}$

10. $-x^2\cos y$

11. $dz/dt = 2x(1)+2y(2t) = 2t+4t^3$

12. $f_{xx}=e^x$, $f_{yy}=0$, $f_{xy}=1$

Multiple Integrals