📖 Calculus II

Module IV: Partial Derivatives

KU2DSCMAT101 | Semester II | DSC Course

Section 14.1 – Functions of Several Variables

Definitions

  • A real-valued function of \(n\) variables: \(w = f(x_1, x_2, \dots, x_n)\)
  • Domain: The largest set of inputs for which the rule gives real numbers.
  • Range: The set of output values.

Domain Examples

FunctionDomainRange
\(z = \sqrt{y - x^2}\)\(y \ge x^2\)\([0, \infty)\)
\(z = \frac{1}{xy}\)\(xy \neq 0\)\((-\infty,0) \cup (0,\infty)\)
\(z = \sin(xy)\)Entire plane\([-1, 1]\)
\(w = \sqrt{x^2 + y^2 + z^2}\)Entire space\([0, \infty)\)

Important Terminology

  • Level curve of \(f(x,y)\): Set of points where \(f(x,y) = c\)
  • Contour curve: Intersection of surface \(z = f(x,y)\) with plane \(z = c\)
  • Level surface of \(f(x,y,z)\): Set of points where \(f(x,y,z) = c\)

Section 14.2 – Limits and Continuity

📖 Two-Path Test for Nonexistence of a Limit
If \(f(x,y)\) has different limits along two different paths as \((x,y) \to (x_0,y_0)\), then \(\lim f(x,y)\) does NOT exist.

Properties of Limits

\(\lim(f+g)=L+M\)   |   \(\lim(f-g)=L-M\)   |   \(\lim(kf)=kL\)
\(\lim(f\cdot g)=L\cdot M\)   |   \(\lim(f/g)=L/M\ (M \neq 0)\)   |   \(\lim[f]^n=L^n\)

Continuity Definition

\(f(x,y)\) is continuous at \((x_0,y_0)\) if:

  1. \(f(x_0,y_0)\) is defined
  2. \(\lim_{(x,y)\to(x_0,y_0)} f(x,y)\) exists
  3. \(\lim f(x,y) = f(x_0,y_0)\)
⭐ Important: Polynomials and rational functions are continuous wherever defined.

Section 14.3 – Partial Derivatives

\(\boxed{\frac{\partial f}{\partial x}\bigg|_{(x_0,y_0)} = \lim_{h\to 0} \frac{f(x_0+h, y_0) - f(x_0, y_0)}{h}}\)
\(\boxed{\frac{\partial f}{\partial y}\bigg|_{(x_0,y_0)} = \lim_{h\to 0} \frac{f(x_0, y_0+h) - f(x_0, y_0)}{h}}\)

Second-Order Partial Derivatives

\(f_{xx} = \frac{\partial^2 f}{\partial x^2},\quad f_{yy} = \frac{\partial^2 f}{\partial y^2},\quad f_{xy} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right),\quad f_{yx} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right)\)
📖 Mixed Derivative Theorem (Clairaut's Theorem)
If \(f, f_x, f_y, f_{xy}, f_{yx}\) are continuous throughout an open region containing \((a,b)\), then \[ \boxed{f_{xy}(a,b) = f_{yx}(a,b)} \] Order of differentiation does not matter if continuity holds.

Implicit Differentiation with Partial Derivatives

\(\boxed{\frac{dy}{dx} = -\frac{F_x}{F_y}}\)    for \(F(x,y)=0\) \((F_y \neq 0)\)
\(\boxed{\frac{\partial z}{\partial x} = -\frac{F_x}{F_z},\quad \frac{\partial z}{\partial y} = -\frac{F_y}{F_z}}\)    for \(F(x,y,z)=0\) \((F_z \neq 0)\)

Section 14.4 – Chain Rule

\(\boxed{\frac{dw}{dt} = \frac{\partial w}{\partial x}\frac{dx}{dt} + \frac{\partial w}{\partial y}\frac{dy}{dt}}\)   (2 intermediate)
\(\boxed{\frac{dw}{dt} = \frac{\partial w}{\partial x}\frac{dx}{dt} + \frac{\partial w}{\partial y}\frac{dy}{dt} + \frac{\partial w}{\partial z}\frac{dz}{dt}}\)   (3 intermediate)
⚠️ Common Pitfalls:
  • Confusing partial derivatives with ordinary derivatives – Use \(\partial\), not \(d\).
  • Assuming limit exists because all straight lines give same value – Curved paths may differ.
  • Forgetting that partial derivatives can exist without continuity.
  • Dividing by zero in implicit differentiation – Always check \(F_z \neq 0\).

❓ Module IV - Questions

1. Determine the domain and range of \(f(x,y) = x^2 + xy^3\). 📖 View Answer
2. Determine the domain and range of \(f(x,y) = \sin(xy)\). 📖 View Answer
3. When do we say that a function \(f(x,y)\) is continuous at \((x_0, y_0)\)? 📖 View Answer
4. Determine the domain and range of \(f(x,y,z) = \sqrt{49 - x^2 - y^2 - z^2}\). 📖 View Answer
5. Determine the domain and range of \(f(x,y) = \sqrt{y - x - 2}\). 📖 View Answer
6. Determine the domain for \(f(x,y) = \ln(x^2 + y^2 - 4)\). 📖 View Answer
7. Determine the domain for \(f(x,y) = \frac{(x-1)(y+2)}{(y-x)(y-x^3)}\). 📖 View Answer
8. Find \(\frac{\partial^2 f}{\partial y \partial x}\) if \(f(x,y) = x\cos y + ye^x\). 📖 View Answer
9. Find \(\frac{\partial^2 f}{\partial x^2}\) if \(f(x,y) = e^{xy}\ln y\). 📖 View Answer
10. State the mixed derivative theorem for a function of two variables. 📖 View Answer
11. Find \(\frac{dy}{dx}\) at \((0,\ln 2)\) if \(xe^x + \sin y + y - \ln 2 = 0\), using partial derivatives. 📖 View Answer
12. Find all second order partial derivatives of \(f(x,y) = x + y + xy\). 📖 View Answer
13. Find all second order partial derivatives of \(f(x,y) = xe^y + y + 1\). 📖 View Answer
14. Find \(f_{yy}\) if \(f(x,y) = \sin(xy)\). 📖 View Answer
15. Find \(f_{xy}\) if \(f(x,y) = \ln(x + y)\). 📖 View Answer
16. Find \(f_{xxxx}\) if \(f(x,y) = xy^2 + x^2y^3 + x^3y^4\). 📖 View Answer
17. Find \(\lim_{(x,y)\to(1,1)}\frac{xy - y - 2x + 2}{x - 1}\). 📖 View Answer
18. Find \(\lim_{(x,y)\to(2,-4)}\frac{y+4}{x^2y - xy + 4x^2 - 4x}\). 📖 View Answer
19. Find \(\lim_{(x,y)\to(4,3)}\frac{\sqrt{x} - \sqrt{y+1}}{x - y - 1}\). 📖 View Answer
20. Show that \(f(x,y) = -\frac{x}{\sqrt{x^2+y^2}}\) has no limit as \((x,y)\to(0,0)\). 📖 View Answer
21. Show that \(f(x,y) = \frac{x^4}{x^4+y^2}\) has no limit as \((x,y)\to(0,0)\). 📖 View Answer
22. Show that \(f(x,y) = \frac{x^4 - y^2}{x^4 + y^2}\) has no limit as \((x,y)\to(0,0)\). 📖 View Answer
23. Show that \(g(x,y) = \frac{x - y}{x + y}\) has no limit as \((x,y)\to(0,0)\). 📖 View Answer
24. Show that \(h(x,y) = \frac{x^2 + y}{y}\) has no limit as \((x,y)\to(0,0)\). 📖 View Answer
25. Show that \(\lim_{(x,y)\to(1,1)}\frac{xy^2-1}{y-1}\) does not exist. 📖 View Answer
26. If \(yz - \ln z = x + y\), find \(\frac{\partial z}{\partial x}\). 📖 View Answer
27. The plane \(x=1\) intersects \(z=x^2+y^2\) in a parabola. Find slope of tangent at \((1,2,5)\). 📖 View Answer
28. Show that \(\lim_{(x,y)\to(1,-1)}\frac{xy+1}{x^2-y^2}\) does not exist. 📖 View Answer
29. For parallel resistors, \(\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}\). Find \(\frac{\partial R}{\partial R_2}\) at \(R_1=30,R_2=45,R_3=90\). 📖 View Answer
30. Show that \(f(x,y) = \frac{2xy}{x^2+y^2}\) (with \(f(0,0)=0\)) is not continuous at origin. 📖 View Answer
31. Show that \(f(x,y) = \frac{2x^2y}{x^4+y^2}\) (with \(f(0,0)=0\)) is not continuous at origin. 📖 View Answer
32. Find the domain of \(f(x,y) = \frac{3x^2y}{x^2+y^2}\). 📖 View Answer
33. Define \(f(0,0)\) to extend \(f(x,y) = \frac{3x^2y}{x^2+y^2}\) continuously. 📖 View Answer
34 Find domain and range of \(f(x,y) = \ln(xy + x - y - 1)\). 📖 View Answer
35 Find domain of \(f(x,y) = \sqrt{(x^2-4)(y^2-9)}\). 📖 View Answer
36 Find domain and range of \(f(x,y) = \frac{1}{\ln(4-x^2-y^2)}\). 📖 View Answer
37. Use chain rule to find \(\frac{dw}{dt}\) for \(w=xy\), \(x=\cos t, y=\sin t\) at \(t=\pi/2\). 📖 View Answer
38. Find \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) at \((0,0,0)\) if \(x^3+z^2+ye^{xz}+z\cos y=0\). 📖 View Answer

✅ Detailed Answers

1. Determine the domain and range of \(f(x,y) = x^2 + xy^3\).
Step 1: Polynomial → Domain = \(\mathbb{R}^2\).
Step 2: Set \(x=1\): \(f=1+y^3\) covers all reals → Range = \(\mathbb{R}\).
Answer: Domain = \(\mathbb{R}^2\), Range = \(\mathbb{R}\)
2. Determine the domain and range of \(f(x,y) = \sin(xy)\).
Step 1: \(\sin\) defined for all reals → Domain = \(\mathbb{R}^2\).
Step 2: \(\sin(\theta) \in [-1,1]\) → Range = \([-1,1]\).
Answer: Domain = \(\mathbb{R}^2\), Range = \([-1,1]\)
3. When do we say that a function \(f(x,y)\) is continuous at \((x_0, y_0)\)?
1. \(f(x_0,y_0)\) is defined
2. \(\lim_{(x,y)\to(x_0,y_0)} f(x,y)\) exists
3. \(\lim f(x,y) = f(x_0,y_0)\)
Answer: All three conditions must be satisfied.
4. Determine the domain and range of \(f(x,y,z) = \sqrt{49 - x^2 - y^2 - z^2}\).
Step 1: \(49 - x^2 - y^2 - z^2 \ge 0 \implies x^2+y^2+z^2 \le 49\).
Step 2: Range = \([0,7]\).
Answer: Domain: \(x^2+y^2+z^2 \le 49\), Range: \([0,7]\)
5. Determine the domain and range of \(f(x,y) = \sqrt{y - x - 2}\).
Step 1: \(y - x - 2 \ge 0 \implies y \ge x+2\).
Step 2: Range = \([0,\infty)\).
Answer: Domain: \(y \ge x+2\), Range: \([0,\infty)\)
6. Determine the domain for \(f(x,y) = \ln(x^2 + y^2 - 4)\).
\(\ln(u)\) defined for \(u>0\): \(x^2+y^2-4 > 0 \implies x^2+y^2 > 4\).
Answer: \(x^2+y^2 > 4\)
7. Determine the domain for \(f(x,y) = \frac{(x-1)(y+2)}{(y-x)(y-x^3)}\).
Denominator ≠ 0: \(y \neq x\) and \(y \neq x^3\).
Answer: All points except \(y=x\) and \(y=x^3\)
8. Find \(\frac{\partial^2 f}{\partial y \partial x}\) if \(f(x,y) = x\cos y + ye^x\).
Step 1: \(f_x = \cos y + ye^x\).
Step 2: \(f_{xy} = -\sin y + e^x\).
Answer: \(-\sin y + e^x\)
9. Find \(\frac{\partial^2 f}{\partial x^2}\) if \(f(x,y) = e^{xy}\ln y\).
Step 1: \(f_x = y e^{xy}\ln y\).
Step 2: \(f_{xx} = y^2 e^{xy}\ln y\).
Answer: \(y^2 e^{xy}\ln y\)
10. State the mixed derivative theorem for a function of two variables.
If \(f,f_x,f_y,f_{xy},f_{yx}\) are continuous near \((a,b)\), then \(f_{xy}(a,b)=f_{yx}(a,b)\).
Answer: \(f_{xy}=f_{yx}\) (order doesn't matter if continuous)
11. Find \(\frac{dy}{dx}\) at \((0,\ln 2)\) if \(xe^x + \sin y + y - \ln 2 = 0\).
Step 1: \(F(x,y)=xe^x+\sin y+y-\ln2\).
Step 2: \(F_x=e^x(1+x)\), \(F_y=\cos y+1\).
Step 3: \(\frac{dy}{dx} = -\frac{F_x}{F_y} = -\frac{e^x(1+x)}{\cos y+1}\).
Step 4: At \((0,\ln2)\): \(-\frac{1}{1+\cos(\ln2)}\).
Answer: \(-\frac{1}{1+\cos(\ln2)}\)
12. Find all second order partial derivatives of \(f(x,y)=x+y+xy\).
Step 1: \(f_x=1+y\), \(f_y=1+x\).
Step 2: \(f_{xx}=0\), \(f_{yy}=0\), \(f_{xy}=1\), \(f_{yx}=1\).
Answer: \(f_{xx}=0,\ f_{yy}=0,\ f_{xy}=1,\ f_{yx}=1\)
13. Find all second order partial derivatives of \(f(x,y)=xe^y+y+1\).
Step 1: \(f_x=e^y\), \(f_y=xe^y+1\).
Step 2: \(f_{xx}=0\), \(f_{yy}=xe^y\), \(f_{xy}=e^y\), \(f_{yx}=e^y\).
Answer: \(f_{xx}=0,\ f_{yy}=xe^y,\ f_{xy}=e^y,\ f_{yx}=e^y\)
14. Find \(f_{yy}\) if \(f(x,y) = \sin(xy)\).
Step 1: \(f_y = x\cos(xy)\).
Step 2: \(f_{yy} = -x^2\sin(xy)\).
Answer: \(-x^2\sin(xy)\)
15. Find \(f_{xy}\) if \(f(x,y) = \ln(x+y)\).
Step 1: \(f_x = \frac{1}{x+y}\).
Step 2: \(f_{xy} = -\frac{1}{(x+y)^2}\).
Answer: \(-\frac{1}{(x+y)^2}\)
16. Find \(f_{xxxx}\) if \(f(x,y)=xy^2+x^2y^3+x^3y^4\).
Step 1: \(f_x = y^2+2xy^3+3x^2y^4\)
Step 2: \(f_{xx}=2y^3+6xy^4\), \(f_{xxx}=6y^4\), \(f_{xxxx}=0\).
Answer: \(0\)
17. Find \(\lim_{(x,y)\to(1,1)}\frac{xy - y - 2x + 2}{x - 1}\).
Numerator = \((x-1)(y-2)\). Simplify: \(y-2\). Limit = \(1-2=-1\).
Answer: \(-1\)
18. Find \(\lim_{(x,y)\to(2,-4)}\frac{y+4}{x^2y - xy + 4x^2 - 4x}\).
Denominator = \(x(x-1)(y+4)\). Simplify: \(\frac{1}{x(x-1)}\). Limit = \(\frac{1}{2(2-1)}=\frac{1}{2}\).
Answer: \(\frac{1}{2}\)
19. Find \(\lim_{(x,y)\to(4,3)}\frac{\sqrt{x} - \sqrt{y+1}}{x - y - 1}\).
Multiply conjugate: \(\frac{x-(y+1)}{(x-y-1)(\sqrt{x}+\sqrt{y+1})} = \frac{1}{\sqrt{x}+\sqrt{y+1}}\). Limit = \(\frac{1}{2+2}=\frac{1}{4}\).
Answer: \(\frac{1}{4}\)
20. Show that \(f(x,y) = -\frac{x}{\sqrt{x^2+y^2}}\) has no limit as \((x,y)\to(0,0)\).
Path \(y=0,x>0\): \(f=-1\). Path \(y=0,x<0\): \(f=1\). Different limits → DNE.
Answer: Limit does not exist.
21. Show that \(f(x,y) = \frac{x^4}{x^4+y^2}\) has no limit as \((x,y)\to(0,0)\).
Path \(y=0\): \(f=1\). Path \(y=x^2\): \(f=\frac{1}{2}\). Different limits → DNE.
Answer: Limit does not exist.
22. Show that \(f(x,y) = \frac{x^4 - y^2}{x^4 + y^2}\) has no limit as \((x,y)\to(0,0)\).
Path \(y=0\): \(f=1\). Path \(x=0\): \(f=-1\). Different limits → DNE.
Answer: Limit does not exist.
23. Show that \(g(x,y) = \frac{x - y}{x + y}\) has no limit as \((x,y)\to(0,0)\).
Path \(y=mx\): \(g=\frac{1-m}{1+m}\) (depends on \(m\)). Different limits → DNE.
Answer: Limit does not exist.
24. Show that \(h(x,y) = \frac{x^2 + y}{y}\) has no limit as \((x,y)\to(0,0)\).
Path \(y=x\): \(h=x+1\to1\). Path \(y=x^2\): \(h=2\). Different limits → DNE.
Answer: Limit does not exist.
25. Show that \(\lim_{(x,y)\to(1,1)}\frac{xy^2-1}{y-1}\) does not exist.
Path \(x=1\): \(\frac{y^2-1}{y-1}=y+1\to2\). Path \(x=2\): \(\frac{2y^2-1}{y-1}\to\infty\). Different limits → DNE.
Answer: Limit does not exist.
26. If \(yz - \ln z = x + y\), find \(\frac{\partial z}{\partial x}\).
\(F=yz-\ln z-x-y=0\). \(F_x=-1\), \(F_z=y-\frac1z\). \(\frac{\partial z}{\partial x}=-\frac{F_x}{F_z}=\frac{1}{y-1/z}=\frac{z}{yz-1}\).
Answer: \(\frac{z}{yz-1}\)
27. The plane \(x=1\) intersects \(z=x^2+y^2\) in a parabola. Find slope of tangent at \((1,2,5)\).
Substitute \(x=1\): \(z=1+y^2\). \(\frac{dz}{dy}=2y\). At \(y=2\): \(4\).
Answer: \(4\)
28. Show that \(\lim_{(x,y)\to(1,-1)}\frac{xy+1}{x^2-y^2}\) does not exist.
Path \(y=-1\): \(\frac{-x+1}{x^2-1}=\frac{-1}{x+1}\to-\frac12\). Path \(x=1\): \(\frac{y+1}{1-y^2}=\frac{1}{1-y}\to\frac12\). Different limits → DNE.
Answer: Limit does not exist.
29. For parallel resistors, find \(\frac{\partial R}{\partial R_2}\) at \(R_1=30,R_2=45,R_3=90\).
\(\frac1R=\frac1{30}+\frac1{45}+\frac1{90}=\frac1{15}\implies R=15\). \(\frac{\partial R}{\partial R_2}=\frac{R^2}{R_2^2}=\left(\frac{15}{45}\right)^2=\frac19\).
Answer: \(\frac19\)
30. Show that \(f(x,y)=\frac{2xy}{x^2+y^2}\) (with \(f(0,0)=0\)) is not continuous at origin.
Path \(y=mx\): \(f=\frac{2m}{1+m^2}\) (depends on \(m\)). Limit DNE → not continuous.
Answer: Not continuous.
31. Show that \(f(x,y)=\frac{2x^2y}{x^4+y^2}\) (with \(f(0,0)=0\)) is not continuous at origin.
Path \(y=x^2\): \(f=1\). Path \(y=0\): \(f=0\). Different limits → not continuous.
Answer: Not continuous.
32. Find the domain of \(f(x,y) = \frac{3x^2y}{x^2+y^2}\).
Denominator \(x^2+y^2=0\) only at \((0,0)\).
Answer: \(\mathbb{R}^2 \setminus \{(0,0)\}\)
33. Define \(f(0,0)\) to extend \(f(x,y) = \frac{3x^2y}{x^2+y^2}\) continuously.
Polar: \(x=r\cos\theta, y=r\sin\theta\) → \(f=3r\cos^2\theta\sin\theta \to 0\).
Answer: \(f(0,0)=0\)
34. Find domain and range of \(f(x,y) = \ln(xy + x - y - 1)\).
\(xy+x-y-1=(x-1)(y+1)>0\) → Case 1: \(x>1,y>-1\); Case 2: \(x<1,y<-1\). Range = \(\mathbb{R}\).
Answer: Domain: \((x>1,y>-1)\cup(x<1,y<-1)\), Range: \(\mathbb{R}\)
35. Find domain of \(f(x,y) = \sqrt{(x^2-4)(y^2-9)}\).
\((x^2-4)(y^2-9)\ge0\) → Both ≥0: \(|x|\ge2,|y|\ge3\); Both ≤0: \(|x|\le2,|y|\le3\).
Answer: \((|x|\ge2,|y|\ge3)\cup(|x|\le2,|y|\le3)\)
36. Find domain and range of \(f(x,y) = \frac{1}{\ln(4-x^2-y^2)}\).
\(4-x^2-y^2>0 \implies x^2+y^2<4\). \(\ln(4-x^2-y^2)\neq0 \implies x^2+y^2\neq3\). Range = \(\mathbb{R}\setminus\{0,\frac{1}{\ln3}\}\).
Answer: Domain: \(x^2+y^2<4,\ x^2+y^2\neq3\), Range: \(\mathbb{R}\setminus\{0,\frac{1}{\ln3}\}\)
37. Use chain rule to find \(\frac{dw}{dt}\) for \(w=xy\), \(x=\cos t, y=\sin t\) at \(t=\pi/2\).
\(\frac{dw}{dt}=y(-\sin t)+x(\cos t)=-\sin^2 t+\cos^2 t=\cos2t\). At \(t=\pi/2\): \(\cos\pi=-1\).
Answer: \(\frac{dw}{dt}=\cos2t\), value = \(-1\)
38. Find \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) at \((0,0,0)\) if \(x^3+z^2+ye^{xz}+z\cos y=0\).
\(F_x=3x^2+yze^{xz}\), \(F_y=e^{xz}-z\sin y\), \(F_z=2z+xye^{xz}+\cos y\). At \((0,0,0)\): \(F_x=0,F_y=1,F_z=1\). \(\frac{\partial z}{\partial x}=-\frac{F_x}{F_z}=0\), \(\frac{\partial z}{\partial y}=-\frac{F_y}{F_z}=-1\).
Answer: \(\frac{\partial z}{\partial x}=0,\ \frac{\partial z}{\partial y}=-1\)
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