Free Printable Partial Derivatives

1.

Find the first order partial derivative with respect to y
$f(x,y)=x^3y^2+3xe^y$
.

a)

^{$f_y(x,y)=3x^2y^2+3e^y$}

b)

$f_y(x,y)=3x^2y^2+2x^3y+3e^y+3xe^y$

c)

$f_y(x,y)=2x^3y+3xe^y$

d)

$f_y(x,y)=6x^2+3e^y$

2.

The temperature at T at a location in the Northern Hemisphere depends on longitude x, latitude y, and time t, so we can write T=f(x,y,t). The time measured since January 1. What is the meaning of the partial derivatives dT/dx?

a)

The rate of change of temperature with respect to time

b)

The rate of change of temperature as longitude and time varies

c)

The rate of change of temperature as longitude varies with latitude and time fixed

d)

The rate of change of the time with respect to temperature

3.

Find $f_x$ and $f_y$ from the equation $f\left(x,y\right)=\ \ln\left(\sqrt{x^2y^3}\right)$

a)

$f_x=\frac{1}{x}\ \ \ \ \ \ \ f_y=\frac{3}{2y}$

b)

$f_x=\frac{1}{xy}\ \ \ \ \ \ \ f_y=\frac{3x}{2y}$

c)

$f_x=\frac{1}{x}\ \ \ \ \ \ \ \ f_y=\frac{3x}{2}$

d)

$f_x=\frac{x^2}{y^3}\ \ \ \ \ f_y=\frac{3x^2}{2y^3}$

4.

Find the first order partial derivative with respect to y
$f(x,y)=x^3y^2+3xe^y$
.

a)

^{$f_y(x,y)=3x^2y^2+3e^y$}

b)

$f_y(x,y)=3x^2y^2+2x^3y+3e^y+3xe^y$

c)

$f_y(x,y)=2x^3y+3xe^y$

d)

$f_y(x,y)=6x^2+3e^y$

5.

Find the Partial derivative of f with respect to y for $f\left(x,y\right)=\frac{y^2}{x+y}$

a)

$\frac{2x+y}{\left(x+y\right)^2}$

b)

$\frac{2xy+y^2}{\left(x+y\right)^2}$

c)

$\frac{2}{\left(x+y\right)^{ }}$

d)

$\frac{2x^2+y^2}{\left(x+y\right)^2}$

6.

$find\ \frac{\partial f}{\partial y},\ given\ f\left(x,y\right)=e^x+2xy^2-4y\ at\ \left(0,3\right)$

a)

$8$

b)

$-4$

c)

$-12$

Partial Derivatives

Total questions: 6

Worksheet time: 11mins

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