4.3 Partial Derivatives

← Chapter 4 §4.3 Partial Derivatives 偏导数 · Chapter 4 · Differentiation of Functions of Several Variables

Access for free at openstax.org. Adapted from Calculus Volume 3 by OpenStax, licensed CC BY-NC-SA 4.0; these lesson pages are shared under the same license. Not affiliated with or endorsed by OpenStax or Rice University.

p1

p330

p331

p332

p333

p334

p335

p336

p337

p338

p339

p340

p341

p342

p343

p344

p345

p346

p347

p862

p867

p868

Access for free at openstax.org · CC BY-NC-SA 4.0

Equation	Name	Unknown
$u_t=c^2(u_{xx}+u_{yy})$	heat equation (2D)	$u(x,y,t)$
$u_{tt}=c^2(u_{xx}+u_{yy})$	wave equation (2D)	$u(x,y,t)$
$u_{xx}+u_{yy}=0$	Laplace's equation (2D)	$u(x,y)$

Objective	Takeaways; say them first, then click
4.3.1 Calculate the partial derivatives of a function of two variables.	limit with one variable frozen; others are constants; slopes of the slices along the axes; estimate from a contour map
4.3.2 Calculate the partial derivatives of a function of more than two variables.	freeze all but one: $f_x,f_y,f_z$
4.3.3 Determine the higher-order derivatives of a function of two variables.	$f_{xx},f_{xy},f_{yx},f_{yy}$ ; $f_{xy}=f_{yx}$ when continuous (Clairaut)
4.3.4 Explain the meaning of a partial differential equation and give an example.	heat, wave, Laplace; verify a solution by substituting its partials

Mistake	Result	Fix
Differentiating the frozen variable	$f_x$ of $2y^2$ written as $4y$	a term without $x$ is a constant: 0
Dropping the inner derivative	$g_x=\cos(x^2y-2x+4)$	chain rule: times $2xy-2$
$f_{xy}$ read as "y first"	order swapped when it matters	subscripts left to right; $\partial^2 f/\partial y\,\partial x$ right to left
Contour estimate taken as exact	$-1.688$ for $-1.118$	it is a secant; differentiate for the tangent
Clairaut used without continuity	$f_{xy}(0,0)=f_{yx}(0,0)$ "proved" for the counterexample	check the hypothesis (p338)
Quotient rule in the wrong order	sign error in Example 4.18a	$(\text{top}'\cdot\text{bottom}-\text{top}\cdot\text{bottom}')/\text{bottom}^2$

#	task	answer
4.12	$4x^2+2xy-y^2+3x-2y+5$ , by limits	$f_x=8x+2y+3$ , $f_y=2x-2y-2$
4.13	$\tan(x^3-3x^2y^2+2y^4)$	$f_x=(3x^2-6xy^2)\sec^2(\cdot)$ , $f_y=(-6x^2y+8y^3)\sec^2(\cdot)$
4.14	$x^2-y^2$ , $f_y$ at $(0,\sqrt2)$ from contours	$\approx-\sqrt3-\sqrt2\approx-3.146$ ; exact $-2\sqrt2\approx-2.828$
4.15	$2x^2-4x^2y+2y^2+5xz^2-6x+3z-8$	$f_x=4x-8xy+5z^2-6$ , $f_y=-4x^2+4y$ , $f_z=10xz+3$
4.16	$\sec(x^2y)-\tan(x^3yz^2)$	$f_z=-2x^3yz\sec^2(x^3yz^2)$ ; $f_x$ , $f_y$ in the key
4.17	$\sin(3x-2y)+\cos(x+4y)$	$f_{xx}=-9\sin(3x-2y)-\cos(x+4y)$ ; $f_{xy}=f_{yx}=6\sin(3x-2y)-4\cos(x+4y)$ ; $f_{yy}=-4\sin(3x-2y)-16\cos(x+4y)$
4.18	$2\sin\tfrac x3\sin\tfrac y4\,e^{-25t/16}$ solves $u_t=9(u_{xx}+u_{yy})$	$u_t=-\tfrac{25}{16}u$ , $9\big(-\tfrac19-\tfrac1{16}\big)u=-\tfrac{25}{16}u$ (not in the key)

#	task	answer	key step
113	$z_y$ , $z=x^2-3xy+y^2$ , by limits	$-3x+2y$	$z(x,y+k)-z=-3xk+2yk+k^2$
115, 117	signs from the bowl's graph	$f_x(-1,1)<0$ ; $f_x(0,0)=0$	slope of the slice
119	$z_y$ , $z=\sin3x\cos3y$	$-3\sin(3x)\sin(3y)$	$\sin3x$ is a constant
121	$\ln(x^6+y^4)$	$\dfrac{6x^5}{x^6+y^4}$ , $\dfrac{4y^3}{x^6+y^4}$	chain rule
125	$\tan(2x-y)$	$2\sec^2(2x-y)$ , $-\sec^2(2x-y)$	inner derivatives 2, −1
127	$\arctan(y/x)$ at $(2,-2)$	$f_x=f_y=\tfrac14$	$f_x=-y/(x^2+y^2)$
129	$z_x$ of $e^{-x}\cos y$ at $(0,1)$	$-\cos1$	—
133	cylinder $V(r,h)$	$\pi r^2h$ ; $V_r=2\pi rh$ ; $V_h=\pi r^2$	circumference × height; base area
135	$f_{xy}$ , $z=\ln(x-y)$	$\dfrac1{(x-y)^2}$	$z_x=(x-y)^{-1}$
143	$f_{xyz}$ , $x^2y^3z-3xy^2z^3+5x^2z-y^3z$	$6xy^2-18yz^2$	one variable at a time
145	$f_x=f_y=0$ , $x^2+x-3xy+y^3-5$	$(\tfrac14,\tfrac12)$ , $(1,1)$	$x=y^2$ , then $2y^2-3y+1=0$
147	$f_x=f_y=0$ , $y^3-3yx^2-3y^2-3x^2+1$	$(0,0),(0,2),(\pm\sqrt3,-1)$	$f_x=-6x(y+1)$
151	$e^{-t}\cos(x/c)$ solves $z_t=c^2z_{xx}$	$c^2z_{xx}=-e^{-t}\cos(x/c)=z_t$	the key omits the minus sign
159	$F=x^2+y^2-2xy\cos\theta$ at $(2,3,\pi/6)$	$F_\theta=6$ , $F_x=4-3\sqrt3$	$F_\theta=2xy\sin\theta$
161	$200x^{0.7}y^{0.3}$ at $(500,1000)$	$f_x\approx172.36$ , $f_y\approx36.93$	$140(y/x)^{0.3}$ , $60(x/y)^{0.7}$

Partial Derivatives

A partial derivative is the ordinary derivative with the other variable held fixed

From the limit, every term without $h$ cancels: $f_x=2x-3y-4$

To find $\partial f/\partial x$ , treat every other variable as a constant

Chain rule: the inner derivative is taken in the moving variable only

$f_x$ and $f_y$ are the slopes of the tangent lines parallel to the $x$ - and $y$ -axes

$f_x(a,b)$ is the slope of the curve cut from the surface by the plane $y=b$

The slope of a slice depends on the point and on the direction

Secant slopes along the slice close in on $f_x$ : $-1,\ -0.68,\ -0.53\to-\tfrac12$

A contour map estimates $g_x$ as rise over run between level curves

For $f_y$ , run along $y$ : $f_y(0,\sqrt2)\approx-3.146$ , exact $-2.828$

A contour map gives a secant; only the limit gives the partial

With three variables, freeze two and differentiate the third

Quotient and chain rules work unchanged; frozen variables ride along as constants

Differentiate a partial again: four second-order partials, two of them mixed

Both orders give the same mixed partial: $f_{xy}=f_{yx}=-3e^{-3y}+10\sin(2x-5y)$

If the mixed partials are continuous, the order of differentiation does not matter

Both orders agree, except where continuity fails

One corner sum equals $h^2f_{xy}$ at one point and $h^2f_{yx}$ at another

Without continuity the mixed partials can differ: $f_{xy}(0,0)=-1$ , $f_{yx}(0,0)=1$

A partial differential equation ties an unknown function of several variables to its partial derivatives

The wave equation checks out: both sides equal $-500\pi^2S$

Every partial is a multiple of $u$ , and the multiples match: $u_t=9(u_{xx}+u_{yy})$

In one dimension, $u_t=c^2u_{xx}$ : the wire's hot middle cools fastest

A wiggly profile has a large $u_{xx}$ , so it fades fast

Higher modes decay like $e^{-\pi^2m^2t}$ : sharp heat dies first

One variable moves, the rest are constants — and every Calc I rule applies

Common mistakes

Practice and answers

Partial Derivatives

A partial derivative is the ordinary derivative with the other variable held fixed

From the limit, every term without hhh cancels: fx=2x−3y−4f_x=2x-3y-4fx​=2x−3y−4

To find ∂f/∂x\partial f/\partial x∂f/∂x, treat every other variable as a constant

Chain rule: the inner derivative is taken in the moving variable only

fxf_xfx​ and fyf_yfy​ are the slopes of the tangent lines parallel to the xxx- and yyy-axes

fx(a,b)f_x(a,b)fx​(a,b) is the slope of the curve cut from the surface by the plane y=by=by=b