Partial Derivatives and Their Applications

The video tutorial explains how to find partial derivatives of functions with three variables. It covers the process of treating other variables as constants while differentiating with respect to one variable. Two examples are provided: one with a simple function f(x, y, z) = 5xyz and another with a more complex function f(x, y, z) = 3x^3.3y^6.2 + z^7. The tutorial demonstrates the step-by-step calculation of partial derivatives for each variable in both examples.

What is the process to find the partial derivative of a function with respect to one variable?

• Treat all other variables as constants and differentiate with respect to the chosen variable.

• Ignore the chosen variable and differentiate with respect to the others.

• Differentiate with respect to all variables simultaneously.

• Integrate the function with respect to the chosen variable.

For the function f(x, y, z) = 5xyz, what is the partial derivative with respect to x?

• 5xyz

• 5yz

• 5xy

• 5xz

When finding the partial derivative of f(x, y, z) = 5xyz with respect to y, what do you treat as constants?

• x and z

• x and y

• y and z

• None

What is the partial derivative of f(x, y, z) = 5xyz with respect to z?

• 5xy

• 5yz

• 5xz

• 5xyz

In the function f(x, y, z) = 3x^3.3y^6.2 + z^7, what is the partial derivative with respect to x?

• 3x^3.3y^6.2

• 9.9x^2.3y^6.2

• 7z^6

• 0

What is the exponent of x in the partial derivative of f(x, y, z) = 3x^3.3y^6.2 + z^7 with respect to x?

• 2.3

• 3.3

• 6.2

• 7

For the function f(x, y, z) = 3x^3.3y^6.2 + z^7, what is the partial derivative with respect to y?

• 3x^3.3y^6.2

• 7z^6

• 0

• 18.6x^3.3y^5.2

What is the coefficient of the partial derivative of f(x, y, z) = 3x^3.3y^6.2 + z^7 with respect to y?

• 18.6

• 9.9

• 7

• 3

What is the partial derivative of f(x, y, z) = 3x^3.3y^6.2 + z^7 with respect to z?

• 0

• 3x^3.3y^6.2

• 7z^6

• 18.6x^3.3y^5.2

In the function f(x, y, z) = 3x^3.3y^6.2 + z^7, what is the exponent of z in the partial derivative with respect to z?

• 3.3

• 5.2

• 7

• 6