Question

Find the first partial derivatives with respect to \(x, y\), and \(z\). $$ w=x^{2}-3 x y+4 y z+z^{3} $$

Short answer

The first order partial derivatives are: \(\frac{\partial w}{\partial x}=2x-3y\), \(\frac{\partial w}{\partial y}=-3x+4z\), \(\frac{\partial w}{\partial z}=4y+3z^{2}\)

Step-by-step solution

Find the partial derivative with respect to x

Differentiate function \(w\) with respect to \(x\) while treating \(y\) and \(z\) as constants. This gives: \(\frac{\partial w}{\partial x}=2x-3y\)

Find the partial derivative with respect to y

Differentiate function \(w\) with respect to \(y\) while treating \(x\) and \(z\) as constants. This gives: \(\frac{\partial w}{\partial y}=-3x+4z\)

Find the partial derivative with respect to z

Differentiate function \(w\) with respect to \(z\) while treating \(x\) and \(y\) as constants. This gives: \(\frac{\partial w}{\partial z}=4y+3z^{2}\)

Key concepts

Multivariable Calculus

Multivariable calculus extends the field of calculus to functions of multiple variables. Calculus, traditionally focused on functions of a single variable, deals with limits, derivatives, integrals, and infinite series. When dealing with functions that depend on more than one variable, such as temperature variation over a geographical area involving both latitude and longitude, or the cost function in economics involving multiple inputs, multivariable calculus becomes the tool of choice.

Multivariable calculus introduces tools like partial derivatives, multiple integrals, and vector calculus, which allow us to analyze and solve problems with multiple variables. A firm grasp of this topic is crucial for fields like physics, engineering, and economics, where one often needs to consider several factors simultaneously.

Derivative with Respect to X

The derivative of a function with respect to a variable, say 'x', is indicative of how the function's value changes as 'x' changes, while other variables are held constant. In multivariable calculus, when working with functions like w(x, y, z), we differentiate with respect to 'x' to get the partial derivative denoted as \( \frac{\partial w}{\partial x} \).

This process is akin to taking the usual derivative of a single-variable function, but with the twist that other variables are treated as non-changing quantities – similar to how you would treat numbers. For our exercise, differentiating the function \( w = x^2 - 3xy + 4yz + z^3 \) with respect to 'x' gives \( \frac{\partial w}{\partial x} = 2x - 3y \) as 'y' and 'z' are treated like constants during this process.

Derivative with Respect to Y

When we shift our attention to partial derivatives with respect to variable 'y', we are interested in how changes in 'y' affect the function's output while keeping every other variable, like 'x' and 'z', constant. In formal terms, we're finding \( \frac{\partial w}{\partial y} \).

This concept is similar to taking a snapshot of the function from the perspective of the y-axis and seeing how the function's value moves along that axis. Applying this concept to our exercise function results in \( \frac{\partial w}{\partial y} = -3x + 4z \) by treating 'x' and 'z' as constants and differentiating the rest with respect to 'y'.

Derivative with Respect to Z

Finally, taking the partial derivative of a function with respect to 'z' tells us how sensitive the function is to changes in 'z', holding 'x' and 'y' steady. This operation gives us \( \frac{\partial w}{\partial z} \) for our function. It's a measure of the instant rate of change along the 'z' direction.

In our exercise, when we differentiate \( w = x^2 - 3xy + 4yz + z^3 \) with respect to 'z', while considering 'x' and 'y' to be fixed, we find that \( \frac{\partial w}{\partial z} = 4y + 3z^2 \) which gives insight into how the function's output is impacted by vertical shifts in the z-axis. Understanding this derivative is essential for predicting the function's behavior in three-dimensional space.