AskSia

Plus

Exercise. Compute the partial derivative of

f(x, y)=x / y

, $f(x, y, z)=x \sin...

Jun 13, 2024

Generated Graph

Solution by Steps

step 1

Compute the partial derivative of

f(x, y) = \frac{x}{y}

with respect to

x

step 2

\frac{\partial}{\partial x} \left( \frac{x}{y} \right) = \frac{1}{y}

step 3

Compute the partial derivative of

f(x, y) = \frac{x}{y}

with respect to

y

step 4

\frac{\partial}{\partial y} \left( \frac{x}{y} \right) = -\frac{x}{y^2}

step 5

Compute the partial derivative of

f(x, y, z) = x \sin(yz)

with respect to

x

step 6

\frac{\partial}{\partial x} \left( x \sin(yz) \right) = \sin(yz)

step 7

Compute the partial derivative of

f(x, y, z) = x \sin(yz)

with respect to

y

step 8

\frac{\partial}{\partial y} \left( x \sin(yz) \right) = xz \cos(yz)

step 9

Compute the partial derivative of

f(x, y, z) = x \sin(yz)

with respect to

z

step 10

\frac{\partial}{\partial z} \left( x \sin(yz) \right) = xy \cos(yz)

Answer

The partial derivatives are: 1.

\frac{\partial}{\partial x} \left( \frac{x}{y} \right) = \frac{1}{y}

\frac{\partial}{\partial y} \left( \frac{x}{y} \right) = -\frac{x}{y^2}

\frac{\partial}{\partial x} \left( x \sin(yz) \right) = \sin(yz)

\frac{\partial}{\partial y} \left( x \sin(yz) \right) = xz \cos(yz)

\frac{\partial}{\partial z} \left( x \sin(yz) \right) = xy \cos(yz)

Key Concept

Partial Derivatives

Explanation

Partial derivatives measure how a function changes as its variables change. They are essential in multivariable calculus for understanding the behavior of functions with more than one variable.