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Steps in Finding Partial Derivatives

How to compute partial derivatives of a function of two or more variables:

  1. Identify which variable you are differentiating with respect to (usually $x$, $y$, or $z$).
  2. Treat all other variables as constants. Only the chosen variable is allowed to change.
  3. Differentiate the function using ordinary differentiation rules (power rule, product rule, chain rule, etc.).
  4. Write the result using partial derivative notation such as $\frac{\partial f}{\partial x}$ or $f_x$.
  5. If needed, repeat the process to find higher-order partial derivatives like $f_{xx}$, $f_{yy}$, or mixed partials such as $f_{xy}$ and $f_{yx}$.

Notation for partial derivatives:

$$f_x = \frac{\partial f}{\partial x}$$
$$f_y = \frac{\partial f}{\partial y}$$
$$f_{xx} = \frac{\partial^2 f}{\partial x^2}$$
$$f_{xy} = \frac{\partial^2 f}{\partial x \partial y}$$

Partial derivatives appear in problems involving surfaces, optimization of functions of two variables, and models where a quantity depends on more than one input.

Basic Partial Derivatives — CE Board

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

For $f_x$: Treat $y$ as a constant, differentiate with respect to $x$.

$f_x = 3x^2 + 4xy$

For $f_y$: Treat $x$ as a constant, differentiate with respect to $y$.

$f_y = 2x^2 - 6y$

Mixed Partial Derivative — CE Board

Find $f_{xy}$ if $f(x,\,y) = x^3y^2 + 2xy$.

Step 1: Find $f_x$ (differentiate with respect to $x$, $y$ constant):

$f_x = 3x^2y^2 + 2y$

Step 2: Differentiate $f_x$ with respect to $y$:

$f_{xy} = \dfrac{\partial}{\partial y}(3x^2y^2 + 2y) = 6x^2y + 2$

Partial Derivatives of Exponential — CE Board

Find $\dfrac{\partial z}{\partial x}$ and $\dfrac{\partial z}{\partial y}$ if $z = e^{xy}$.

$\dfrac{\partial z}{\partial x} = y\,e^{xy}$
$\dfrac{\partial z}{\partial y} = x\,e^{xy}$

In both cases the chain rule is applied; the other variable acts as a constant multiplier in the exponent.

All Second Partial Derivatives — CE Board

If $f(x,\,y) = x^2 + 3xy + y^2$, find $f_{xx}$, $f_{yy}$, $f_{xy}$, and $f_{yx}$.

First partial derivatives:

$f_x = 2x + 3y,\quad f_y = 3x + 2y$

Second partial derivatives:

$f_{xx} = 2,\quad f_{yy} = 2,\quad f_{xy} = 3,\quad f_{yx} = 3$

Note: $f_{xy} = f_{yx}$ (Clairaut's theorem — mixed partials are equal for smooth functions).

Applied Partial Derivative — Cylinder (CE Board)

The volume of a cylinder is $V = \pi r^2 h$. Find $\dfrac{\partial V}{\partial r}$ and $\dfrac{\partial V}{\partial h}$, and interpret each result.

$\dfrac{\partial V}{\partial r} = 2\pi r h$

This is the rate of change of volume with respect to radius (height held constant).

$\dfrac{\partial V}{\partial h} = \pi r^2$

This is the rate of change of volume with respect to height (radius held constant) — equal to the base area of the cylinder.

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Exam Generator Problems

Additional board-style practice items for this topic.

Question Bank: q679

MSTE - Differential Calculus / Worded Problems / Engr. Janclyde Espinosa (Clidez)

How many odd three-digit integers greater than 800 are there such that all their digits are different?

  1. 72
  2. 40
  3. 56
  4. 81
Odd three-digit integers greater than 800 have hundreds digit 8 or 9. Case 1, hundreds digit 8: units digit can be 1,3,5,7,9 (5 ways), tens digit then has 8 choices, giving 40. Case 2, hundreds digit 9: units digit can be 1,3,5,7 (4 ways), tens digit has 8 choices, giving 32.
Total $=40+32$.
$\boxed{72}$

Question Bank: q696

MSTE - Differential Calculus / Directional Derivative / Engr. Janclyde Espinosa (Clidez)

Find the gradient of f(x,y) = y ln x + xy2 at the point (1,2). Gradient is defined as ∇f(x,y) = fx(x,y)i + fy(x,y)j where fx(x,y) is the partial derivative with respect to x and fy(x,y) is the partial derivative with respect to y.

How much work is done winding up the upper half of the chain?

  1. 6i + 4j
  2. 6i − 4j
  3. 4i + 6j
  4. 4i − 6j
For $f(x,y)=y\ln x+xy^2$:
$f_x=\frac{y}{x}+y^2$
$f_y=\ln x+2xy$
At $(1,2)$:
$f_x=2+4=6$, $f_y=0+4=4$
$\boxed{\nabla f=6i+4j}$

Question Bank: t812

MSTE - Differential Calculus / Differential Calculus / Gemini mapped Chapter 4 to 6

Two automobiles are approaching the origin. The first one is traveling from the left on the x-axis at 20 kph. The second is traveling from the top on the y-axis at 45 kph. How fast (in kph) is the distance between them changing when the first is at (-5, 0) and the second is at (0, 10)? (Both coordinates are in kilometers).

  1. 49.19 approaching
  2. 49.19 separating
  3. 63.25 separating
  4. 63.25 approaching
Let car 1 be at $x$ (with $\frac{dx}{dt} = +20$, moving toward the origin) and car 2 at $y$ ($\frac{dy}{dt} = -45$).
At $(-5, 0)$ and $(0, 10)$: $s = \sqrt{25 + 100} = 11.18$ km.
$\frac{ds}{dt} = \frac{x\frac{dx}{dt} + y\frac{dy}{dt}}{s} = \frac{(-5)(20) + (10)(-45)}{11.18} = \frac{-550}{11.18}$
The negative sign means the distance is shrinking:
$\boxed{49.19 \text{ kph, approaching}}$

Question Bank: t814

MSTE - Differential Calculus / Differential Calculus / Gemini mapped Chapter 4 to 6

One ship is sailing south at a rate of 5 knots, and another is sailing east at the rate of 10 knots. At 2 P.M., the second ship was at the place occupied by the first ship one hour before.

How far apart are the ships at 12 noon in nautical miles?

  1. 18.63
  2. 21.45
  3. 20.62
  4. 22.32

At what time will the two ships nearest each other?

  1. 1:18 pm
  2. 1:32 pm
  3. 1:57 pm
  4. 1:48 pm

How fast are the ships separating at 3 pm, in knots?

  1. 13.68
  2. 11.25
  3. 10.61
  4. 12.32

Part 1.

Let $t$ be hours after 1 PM, with ship 1 at $(0, -5t)$ and ship 2 (reaching the origin at 2 PM) at $(10(t-1), 0)$.
At noon, $t = -1$: ship 1 at $(0, 5)$, ship 2 at $(-20, 0)$.
$s = \sqrt{20^2 + 5^2} = \sqrt{425}$
$\boxed{20.62 \text{ NM}}$

Part 2.

Distance squared: $s^2 = (10(t-1))^2 + (5t)^2 = 125t^2 - 200t + 100$.
Minimize: $\frac{d(s^2)}{dt} = 250t - 200 = 0 \Rightarrow t = 0.8$ h after 1 PM.
$0.8\text{ h} = 48$ min:
$\boxed{1{:}48 \text{ PM}}$

Part 3.

At 3 PM, $t = 2$: $s^2 = 125(4) - 200(2) + 100 = 200$, so $s = 14.14$ NM.
$2s\frac{ds}{dt} = 250t - 200 = 250(2) - 200 = 300$
$\frac{ds}{dt} = \frac{300}{2(14.14)}$
$\boxed{10.61 \text{ knots}}$

Question Bank: t865

MSTE - Differential Calculus / Differential Calculus / Gemini mapped Chapter 4 to 6

Find the length of the arc of $x^2 + y^2 = 64$ from $x = -1$ to $x = -3$, in the second quadrant.

  1. 2.24
  2. 2.61
  3. 2.75
  4. 2.07
On the circle $r = 8$, arc length $= r\,\Delta\theta$. Find the central angles at the endpoints:
At $x = -1$: $\theta_1 = \cos^{-1}\!\frac{-1}{8} = 97.18^\circ$
At $x = -3$: $\theta_2 = \cos^{-1}\!\frac{-3}{8} = 112.02^\circ$
$\Delta\theta = 14.84^\circ = 0.259$ rad
$s = 8(0.259)$
$\boxed{s = 2.07}$

Question Bank: t1412

MSTE - Differential Calculus / Derivatives / BEMz

Find the derivative of $y=2^{4x}$.

  1. $3^{4x+2} \ln 2$
  2. $2^{4x+2} \ln 2$
  3. $6^{3x+2} \ln 2$
  4. $4^{4x+2} \ln 2$

Solution pending in psadquestions/t1412.json.

Question Bank: t1413

MSTE - Differential Calculus / Derivatives / BEMz

Find the derivative of h with respect to u if $h=\pi^{2u}$.

  1. $\pi^{2u}$
  2. $2u \ln \pi$
  3. $2\pi^{2u} \ln \pi$
  4. $2\pi^{2u}$

Solution pending in psadquestions/t1413.json.

Question Bank: t1416

MSTE - Differential Calculus / Derivatives / BEMz

Find the derivative of log a u with respect to x.

  1. $\log u du/dx$
  2. $u du/\ln a$
  3. $\log a e/u$
  4. $\log a du/dx$

Solution pending in psadquestions/t1416.json.

Question Bank: t1417

MSTE - Differential Calculus / Derivatives / BEMz

Find the derivative of arc cos (2x).

  1. $- 2/\sqrt{1 -4x^{2}}$
  2. $2/\sqrt{1 -4x^{2}}$
  3. $2/(1+4x^{2})$
  4. $2/\sqrt{2x^{2} -1}$

Solution pending in psadquestions/t1417.json.

Question Bank: t1418

MSTE - Differential Calculus / Derivatives / BEMz

Find the derivative of 4 arc tan (2x).

  1. $4/(1+x^{2})$
  2. $4/(4x^{2}+1)$
  3. $8/(1+4x^{2})$
  4. $8/(4x^{2}+1)$

Solution pending in psadquestions/t1418.json.

Question Bank: t1419

MSTE - Differential Calculus / Derivatives / BEMz

Find the derivative of arc csc (3x).

  1. $- 1/[x\sqrt{9x^{2} -1}]$
  2. $1/[3x\sqrt{9x^{2} -1}]$
  3. $3/[x\sqrt{1 -9x^{2}}]$
  4. $3/[x\sqrt{9x}^{2} -1)]$

Solution pending in psadquestions/t1419.json.

Question Bank: t1420

MSTE - Differential Calculus / Derivatives / BEMz

Find the derivative of arc sec (2x)

  1. $1/[x\sqrt{4x^{2} -1}]$
  2. $2/[x\sqrt{4x^{2} -1}]$
  3. $1/[x\sqrt{1 -4x^{2}}]$
  4. $2/[x\sqrt{1 -4x^{2}}]$

Solution pending in psadquestions/t1420.json.

Question Bank: t1423

MSTE - Differential Calculus / Derivatives / BEMz

Find the derivative of y with respect to x if y = x ln x – x.

  1. $x \ln x$
  2. $\ln x$
  3. $(\ln x)/x$
  4. $x/\ln x$

Solution pending in psadquestions/t1423.json.

Question Bank: t1425

MSTE - Differential Calculus / Derivatives / BEMz

Find the derivative of $y=x^{x}$.

  1. $x^{x} (2+\ln x)$
  2. $x^{x} (1+\ln x)$
  3. $x^{x} (4-\ln x)$
  4. $x^{x} (8+\ln x)$

Solution pending in psadquestions/t1425.json.

Question Bank: t1426

MSTE - Differential Calculus / Derivatives / BEMz

Find the derivative of $y=\log$ a 4x.

  1. $y’=(\log a e)/x$
  2. $y’=(\cos e)/x$
  3. $y’=(\sin e)/x$
  4. $y’=(\tan e)/x$

Solution pending in psadquestions/t1426.json.

Question Bank: t1427

MSTE - Differential Calculus / Derivatives / BEMz

What is the derivative with respect to x of $(x+1)^{3}$ – $x^{3}$.

  1. 3x+3
  2. 3x-3
  3. 6x-3
  4. 6x+3

Solution pending in psadquestions/t1427.json.

Question Bank: t1428

MSTE - Differential Calculus / Derivatives / BEMz

What is the derivative with respect to x of $\sec^{2}$ (x)?

  1. $2x \sec^{2} (x) \tan^{2} (x)$
  2. $2x \sec (x) \tan (x)$
  3. $\sec^{2} (x) \tan^{2} (x)$
  4. $2 \sec^{2} (x) \tan^{2} (x)$

Solution pending in psadquestions/t1428.json.

Question Bank: t1429

MSTE - Differential Calculus / Derivatives / BEMz

The derivative with respect to x of $2\cos^{2} (x^{2}+2)$.

  1. $4 \sin (x^{2}+2) \cos (x^{2}+2)$
  2. $-4 \sin (x^{2}+2) \cos (x^{2}+2)$
  3. $8x \sin (x^{2}+2) \cos (x^{2}+2)$
  4. $-8x \sin (x^{2}+2) \cos (x^{2}+2)$

Solution pending in psadquestions/t1429.json.

Question Bank: t1430

MSTE - Differential Calculus / Derivatives / BEMz

Find the derivative of $[(x+1)^{3}]/x$.

  1. $[3(x+1)^{2}]/x - [(x+1)^{3}]/x^{2}$
  2. $[2(x+1)^{3}]/x - [(x+1)^{3}]/x^{3}$
  3. $[4(x+1)^{2}]/x - [2(x+1)^{3}]/x$
  4. $[(x+1)^{2}]/x - [(x+1)^{3}]/x$

Solution pending in psadquestions/t1430.json.

Question Bank: t1434

MSTE - Differential Calculus / Derivatives / BEMz

What is the first derivative dy/dx of the expression $(xy)^{x}=e$.

  1. $-y(1-\ln xy)/x^{2}$
  2. $-y(1+\ln xy)/x$
  3. 0
  4. x/y

Solution pending in psadquestions/t1434.json.

Question Bank: t1454

MSTE - Differential Calculus / Derivatives / BEMz

Find the second derivative of $y=(2x+1)^{2}+x^{3}$.

  1. 8+6x
  2. $(2x+1)^{3}$
  3. x+1
  4. 6+4x

Solution pending in psadquestions/t1454.json.

Question Bank: t1455

MSTE - Differential Calculus / Derivatives / BEMz

Find the second derivative of $y=(2x+4)^{2} x^{3}$.

  1. $x^{2}(80x+192)$
  2. 2x+4
  3. $x^{3}(2x+80)$
  4. $x^{2}(20x+60)$

Solution pending in psadquestions/t1455.json.

Question Bank: t1456

MSTE - Differential Calculus / Derivatives / BEMz

Find the second derivative of $y=2x+3(4x+2)^{3}$ when $x=1$.

  1. 1728
  2. 1642
  3. 1541
  4. 1832

Solution pending in psadquestions/t1456.json.

Question Bank: t1457

MSTE - Differential Calculus / Derivatives / BEMz

Find the second derivative of $y=2x/[3(4x+2)^{2}]$ when $x=0$.

  1. -1.33
  2. 1.44
  3. 2.16
  4. -2.72

Solution pending in psadquestions/t1457.json.

Question Bank: t1458

MSTE - Differential Calculus / Derivatives / BEMz

Find the second derivative of $y=3/(4x^{-3})$ when $x=1$.

  1. 4.5
  2. -3.6
  3. 2.4
  4. -1.84

Solution pending in psadquestions/t1458.json.

Question Bank: t1459

MSTE - Differential Calculus / Derivatives / BEMz

Find the second derivative of $y=x^{-2}$ when $x=2$.

  1. 0.375
  2. 0.268
  3. 0.148
  4. 0.425

Solution pending in psadquestions/t1459.json.

Question Bank: t1460

MSTE - Differential Calculus / Derivatives / BEMz

Find the first derivative of $y=2\cos(2+x^{2})$.

  1. $-4x \sin (2+x^{2})$
  2. $4x \cos (2+x^{2})$
  3. $x \sin (2+x^{2})$
  4. $x \cos (2+x^{2})$

Solution pending in psadquestions/t1460.json.

Question Bank: t1461

MSTE - Differential Calculus / Derivatives / BEMz

Find the first derivative of $y=2 \sin^{2} (3x^{2}-3)$.

  1. $24x \sin (3x^{2}-3) \cos (3x^{2}-3)$
  2. $12 \sin (3x^{2}-3)$
  3. $6x \cos (3x^{2}-3)$
  4. $24x \sin (3x^{2}-3)$

Solution pending in psadquestions/t1461.json.

Question Bank: t1462

MSTE - Differential Calculus / Derivatives / BEMz

Find the first derivative of $y=\tan^{2} (3x^{2}-4)$.

  1. $12xtan(3x^{2}-4)\sec^{2}(3x^{2}-4)$
  2. $x \tan (3x^{2}-4)$
  3. $\sec^{2} (3x^{2}-4)$
  4. $2 \tan^{2}(3x^{2}-4)\csc^{2}(3x^{2}-4)$

Solution pending in psadquestions/t1462.json.

Question Bank: t1463

MSTE - Differential Calculus / Derivatives / BEMz

Find the derivative of arc cos 4x

  1. $-4/(1-16x^{2})^{0}.5$
  2. $4/(1-16x^{2})^{0}.5$
  3. $-4/(1-4x^{2})^{0}.5$
  4. $4/(1-4x^{2})^{0}.5$

Solution pending in psadquestions/t1463.json.