Partial derivatives practice problems
Differentiate in one variable while holding the others fixed. 23 problems with worked solutions; in 23 of them every equation is proved by a computer algebra system.
For \( \displaystyle f(x, y) = \ln{\left(x^{2} + y^{2} + 1 \right)} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).
For \( \displaystyle f(x, y) = x^{3} - 2 x y + y^{2} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).
For \( \displaystyle f(x, y) = \frac{x + y}{x - y + 3} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).
For \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2} + 4} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).
For \( \displaystyle f(x, y) = e^{x y} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).
For \( \displaystyle f(x, y) = x \sin{\left(y \right)} + y \cos{\left(x \right)} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).
For \( \displaystyle f(x, y) = x^{2} y + 3 x y^{3} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).
For \( \displaystyle f(x, y) = x e^{2 y} + y^{2} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).
For \( \displaystyle f(x, y) = 5 x^{2} y - 4 x y^{3} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).
For \( \displaystyle f(x, y) = 4 x^{2} y - 3 x y^{3} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).
For \( \displaystyle f(x, y) = \frac{x + 3 y}{x - y + 5} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).
For \( \displaystyle f(x, y) = \ln{\left(x^{2} + 3 y^{2} + 1 \right)} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).
For \( \displaystyle f(x, y) = 2 \operatorname{atan}{\left(\frac{y}{x} \right)} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).
For \( \displaystyle f(x, y) = 3 x^{2} y - 4 x y^{3} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).
For \( \displaystyle f(x, y) = \frac{x + 3 y}{x - y + 2} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).
For \( \displaystyle f(x, y) = 2 x^{2} y + 4 x y^{3} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).
For \( \displaystyle f(x, y) = x^{3} - 3 x y + 3 y^{2} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).
For \( \displaystyle f(x, y) = \ln{\left(x^{2} + 2 y^{2} + 1 \right)} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).
For \( \displaystyle f(x, y) = e^{2 x y} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).
For \( \displaystyle f(x, y) = x^{3} - 3 x y + 2 y^{2} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).
For \( \displaystyle f(x, y) = 4 x^{2} y - 4 x y^{3} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).
For \( \displaystyle f(x, y) = x^{3} + x y + 3 y^{2} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).
For \( \displaystyle f(x, y) = x^{2} \ln{\left(y + 1 \right)} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).