Questions: Introduction to partial differentiation
Before attempting these questions, it is highly recommended that you read Guide: Introduction to partial differentiation.
Q1
Find all possible first-order partial derivatives for each function \(f\).
1.1. \(\displaystyle \quad f(x,y) = x^2y + y^3\)
1.2. \(\displaystyle \quad f(x,y) = 3x^3 - 2y^4 + xy\)
1.3. \(\displaystyle \quad f(x,y) = y\sin(2x) + 3\)
1.4. \(\displaystyle \quad f(x,y) = e^{xy} + 2x^2y^3\)
1.5. \(\displaystyle \quad f(x,y) = \ln(x) + x\ln(y) + 3x\)
1.6. \(\displaystyle \quad f(x,y) = \dfrac{y}{x} - \dfrac{x}{y}\)
1.7. \(\displaystyle \quad f(x,y) = x \exp(y^2)\)
1.8. \(\displaystyle \quad f(x,y) = \sqrt{x^2+y^2}\)
1.9. \(\displaystyle \quad f(x,y) = (3x+2y)^4\)
1.10. \(\displaystyle \quad f(x,y) = y \sin(xy)\)
1.11. \(\displaystyle \quad f(x,y) = \sin(x^2+y^2)\)
1.12. \(\displaystyle \quad f(x,y) = \ln(1+x^2y^2)\)
1.13. \(\displaystyle \quad f(x,y,z) = x^2y \sin(z)\)
1.14. \(\displaystyle \quad f(x,y,z) = (x+y)(y+z)(z+x)\)
1.15. \(\displaystyle \quad f(x,y,z) = \dfrac{xyz}{x+y+z}\)
Q2
A function \(f(x,y)\) is called harmonic if it satisfies the equation
\[\dfrac{\partial^2 f}{\partial x^2} + \dfrac{\partial^2 f}{\partial y^2} = 0\]
Show that each of these functions is harmonic by calculating the pure second-order partial derivatives and checking that their sum is zero.
2.1. \(\displaystyle \quad f(x,y) = x^2 - y^2\)
2.2. \(\displaystyle \quad f(x,y) = xy\)
2.3. \(\displaystyle \quad f(x,y) = x^3-3xy^2\)
2.4. \(\displaystyle \quad f(x,y) = \cos(x) \sinh(y)\)
2.5. \(\displaystyle \quad f(x,y) = e^x \sin(y)\)
2.6. \(\displaystyle \quad f(x,y) = \tan^{-1} \left( \dfrac{y}{x} \right)\)
2.7. \(\displaystyle \quad f(x,y) = \ln(x^2+y^2)\)
Q3
For each function \(f(x,y)\), calculate the mixed second-order partial derivatives and confirm that they satisfy the equation
\[\dfrac{\partial^2 f}{\partial x \partial y} = \dfrac{\partial^2 f}{\partial y \partial x}\]
3.1. \(\displaystyle \quad f(x,y) = x^2y + xy^2\)
3.2. \(\displaystyle \quad f(x,y) = 2x^2 \cos(y)\)
3.3. \(\displaystyle \quad f(x,y) = (x+y)^5\)
3.4. \(\displaystyle \quad f(x,y) = \dfrac{x}{1+y}\)
3.5. \(\displaystyle \quad f(x,y) = \sqrt{x^2+y^2}\)
3.6. \(\displaystyle \quad f(x,y) = x^2\sin(y) + y^2\cos(x)\)
3.7. \(\displaystyle \quad f(x,y) = \tan^{-1}(xy)\)
After attempting the questions above, please click this link to find the answers.
Version history and licensing
v1.0: initial version created 05/25 by Donald Campbell as part of a University of St Andrews VIP project.