Multivariate Calculus

J. Zachary Klingensmith

7 Multivariate Calculus

7.1 Partial derivatives

Practice Problems

Problem 7.1.1: Find the partial derivative with respect to each variable in the problem below:

$z=x^2+y^2$

Answer:

$\frac{\partial z}{\partial x}=2x, \frac{\partial z}{\partial y}=2y$

Problem 7.1.2: Find the partial derivative with respect to each variable in the problem below:

$z=x^2-5xy-3y^2$

Answer:

$\frac{\partial z}{\partial x}=2x-5y, \frac{\partial z}{\partial y}=-5x-6y$

Problem 7.1.3: Find the partial derivative with respect to each variable in the problem below:

$f(x,y)=\frac{x^2y}{y-1}$

Answer:

$\frac{\partial f}{\partial x}=\frac{2xy}{y-1}, \frac{\partial f}{\partial y}=\frac{-x^2}{(y-1)^2}$

Problem 7.1.4: Find the partial derivative with respect to each variable in the problem below:

$f(x,y)=(x^2-2xy)^5$

Answer:

$\frac{\partial f}{\partial x}=5(x^2-2xy)^4(2x-2y), \frac{\partial f}{\partial y}=-10x(x^2-2xy)^4$

Problem 7.1.5: Find the partial derivative with respect to each variable in the problem below:

$f(x,y,z)=x^2-2y^2+z^2-2xy+3xz-2yz+10xyz-5$

Answer:

$\frac{\partial f}{\partial x}=2x-2y+3z+10yz, \frac{\partial f}{\partial y}=-4y-2x-2z+10xz, \frac{\partial f}{\partial z}=2z+3x-2y+10xy$

Problem 7.1.6: Find the partial derivative with respect to each variable in the problem below:

$f(x,y,z)=x^2y-xyz^2+3xz^3-4xy^2+2xz^5$

Answer:

$\frac{\partial f}{\partial x}=2xy-yz^2+3z^2-4y^2+2z^5, \frac{\partial f}{\partial y}=x^2+xz^2-8xy, \frac{\partial f}{\partial z}=-2xyz+9xz^2+10xz^4$

Problem 7.1.7: Find the partial derivative with respect to each variable in the problem below:

$f(x,y,z)=(x^2yz-xz^3-xy^2z^2)^8$

Answer:

$\frac{\partial f}{\partial x}=8(x^2yz-xz^3-xy^2z^2)^7(2xyz-z^3-y^2z^2), \frac{\partial f}{\partial y}=8(x^2yz-xz^3-xy^2z^2)^7(x^2z-2xyz^2), \frac{\partial f}{\partial z}=8(x^2yz-xz^3-xy^2z^2)^7(x^2y-3xz^2-2xy^2z)$

Problem 7.1.8: The production function for an economy is given below. Find the equation for the marginal productivity of labor and the marginal productivity of capital (MP_L and MP_K).

$Q=30L^{.8}K^{.2}$

Answers:

$MP_{L}=\frac{\partial Q}{\partial L}=24\left(\frac{K}{L}\right)^{.2}, MP_{K}=\frac{\partial Q}{\partial K}=6\left(\frac{L}{K}\right)^{.8}$

Problem 7.1.9: An individual has the choice to consume different quantities of three goods. The utility function is given below. Find the marginal utility equations for each good.

$U(x_1,x_2,x_3)=(5x_1+1)^3(2x_2+4)^2(5x_3+4)^5$

Answers:

$MU_{1}=\frac{\partial U}{\partial x_{1}}=15(5x_{1}+1)^2(2x_2+4)^2(5x_3+4)^5,MU_{2}=\frac{\partial U}{\partial x_{2}}=4(5x_1+1)^3(2x_2+4)(5x_3+4)^5, MU_{3}=\frac{\partial U}{\partial x_{3}}=25(5x_1+1)^3(2x_2+4)^2(5x_3+4)^4$