Finding Partial Derivative with Respect to s

Oct 13, 2025 by magicalceylontours.com 142 views

Chain Rule in Multivariable Calculus: Finding Partial Derivatives

In multivariable calculus, the chain rule is a powerful tool used to find partial derivatives of composite functions. Given a function of multiple variables, the chain rule allows us to find the partial derivative of the function with respect to one of the variables, while treating the other variables as constants. In this article, we will use the chain rule to find the indicated partial derivatives of a given composite function.

The chain rule states that if we have a composite function of the form:

${ z = f(x, y, ...) }$

where x, y, ... are functions of other variables s, t, u, ..., then the partial derivative of z with respect to s is given by:

${ \frac{\partial z}{\partial s} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial s} + ... }$

Similarly, the partial derivative of z with respect to t is given by:

${ \frac{\partial z}{\partial t} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial t} + ... }$

And so on.

The given function is:

${ z = x^4 + x^2 y, \quad x = s + 2t - u, \quad y = stu^2 }$

We are asked to find the partial derivatives of z with respect to s, t, and u.

To find the partial derivatives of z with respect to s, t, and u, we will use the chain rule.

Finding Partial Derivative with Respect to s

First, we need to find the partial derivatives of z with respect to x and y.

${ \frac{\partial z}{\partial x} = 4x^3 + 2xy }$

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

Next, we need to find the partial derivatives of x and y with respect to s.

${ \frac{\partial x}{\partial s} = 1 }$

${ \frac{\partial y}{\partial s} = tu^2 }$

Now, we can use the chain rule to find the partial derivative of z with respect to s.

${ \frac{\partial z}{\partial s} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial s} }$

${ \frac{\partial z}{\partial s} = (4x^3 + 2xy)(1) + (x^2)(tu^2) }$

${ \frac{\partial z}{\partial s} = 4x^3 + 2xy + x^2 tu^2 }$

Finding Partial Derivative with Respect to t

First, we need to find the partial derivatives of z with respect to x and y.

${ \frac{\partial z}{\partial x} = 4x^3 + 2xy }$

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

Next, we need to find the partial derivatives of x and y with respect to t.

${ \frac{\partial x}{\partial t} = 2 }$

${ \frac{\partial y}{\partial t} = su^2 }$

Now, we can use the chain rule to find the partial derivative of z with respect to t.

${ \frac{\partial z}{\partial t} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial t} }$

${ \frac{\partial z}{\partial t} = (4x^3 + 2xy)(2) + (x^2)(su^2) }$

${ \frac{\partial z}{\partial t} = 8x^3 + 4xy + x^2 su^2 }$

Finding Partial Derivative with Respect to u

First, we need to find the partial derivatives of z with respect to x and y.

${ \frac{\partial z}{\partial x} = 4x^3 + 2xy }$

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

Next, we need to find the partial derivatives of x and y with respect to u.

${ \frac{\partial x}{\partial u} = -1 }$

${ \frac{\partial y}{\partial u} = 2stu }$

Now, we can use the chain rule to find the partial derivative of z with respect to u.

${ \frac{\partial z}{\partial u} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial u} }$

${ \frac{\partial z}{\partial u} = (4x^3 + 2xy)(-1) + (x^2)(2stu) }$

${ \frac{\partial z}{\partial u} = -4x^3 - 2xy + 2x^2 stu }$

In this article, we used the chain rule to find the partial derivatives of a given composite function. We found the partial derivatives of z with respect to s, t, and u, and showed how to use the chain rule to find these derivatives. The chain rule is a powerful tool in multivariable calculus, and is used to find partial derivatives of composite functions.
Chain Rule in Multivariable Calculus: Q&A

In our previous article, we used the chain rule to find the partial derivatives of a given composite function. In this article, we will answer some common questions related to the chain rule in multivariable calculus.

A: The chain rule is a mathematical formula used to find the partial derivatives of a composite function. It states that if we have a composite function of the form:

${ z = f(x, y, ...) }$

where x, y, ... are functions of other variables s, t, u, ..., then the partial derivative of z with respect to s is given by:

${ \frac{\partial z}{\partial s} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial s} + ... }$

A: The chain rule is used to find the partial derivatives of composite functions. A composite function is a function that is composed of other functions. For example, if we have a function of the form:

${ z = f(x, y) }$

where x and y are functions of other variables s, t, u, ..., then we can use the chain rule to find the partial derivative of z with respect to s.

A: To apply the chain rule, we need to follow these steps:

Find the partial derivatives of the inner functions (x, y, ...) with respect to the outer variables (s, t, u, ...).
Find the partial derivatives of the outer function (z) with respect to the inner functions (x, y, ...).
Use the chain rule formula to find the partial derivative of z with respect to the outer variables (s, t, u, ...).

A: Some common mistakes to avoid when using the chain rule include:

Not finding the partial derivatives of the inner functions with respect to the outer variables.
Not finding the partial derivatives of the outer function with respect to the inner functions.
Not using the correct formula for the chain rule.

A: Yes, the chain rule can be used to find the total derivative of a composite function. The total derivative is a measure of the rate of change of the function with respect to all the variables.

A: The chain rule has many real-world applications, including:

Physics: The chain rule is used to find the partial derivatives of physical quantities such as velocity, acceleration, and force.
Engineering: The chain rule is used to find the partial derivatives of engineering quantities such as stress, strain, and temperature.
Economics: The chain rule is used to find the partial derivatives of economic quantities such as price, quantity, and demand.

In this article, we answered some common questions related to the chain rule in multivariable calculus. We hope that this article has been helpful in understanding the chain rule and its applications. If you have any further questions, please don't hesitate to ask.