Problem 30 - Calculus an Applied Approach

Question

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

Step-by-Step Solution

Answer

The first 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 1: Derivative with respect to \(x\)

When other variables \((y,z)\) are treated as constants, the derivative of \(w\) with respect to \(x\) is \[\frac{\partial w}{\partial x} = 2x - 3y.\]

Step 2: Derivative with respect to \(y\)

When other variables \((x,z)\) are treated as constants, the derivative of \(w\) with respect to \(y\) is \[\frac{\partial w}{\partial y} = -3x+4z.\]

Step 3: Derivative with respect to \(z\)

When other variables \((x,y)\) are treated as constants, the derivative of \(w\) with respect to \(z\) is \[\frac{\partial w}{\partial z} = 4y+3z^{2}.\]

Key Concepts

Multivariable Calculus

Multivariable Calculus is a branch of mathematics that extends the concepts of calculus to functions of more than one variable. Unlike single-variable calculus which deals with functions of one variable, multivariable calculus covers functions like \(w(x, y, z)\) where multiple inputs affect the output. This area of calculus is crucial for modeling and understanding systems in physics, engineering, economics, and more complex real-world situations.

• In multivariable calculus, variables mutually affect each other and the outcome, necessitating a nuanced approach to function analysis.


• The primary tools of analysis here include gradients, partial derivatives, and multiple integrals.

Working with multiple variables means understanding how changes in one variable affect the outcome when others are constant. This concept is central to many applications like optimization and finding surface tangents in physical sciences.

Derivative with respect to a variable

The derivative with respect to a variable measures how a function changes as that specific variable changes, while other variables are held constant. In the context of functions of several variables, this is where partial derivatives come into play.

• For a function \(w(x, y, z)\), the derivative with respect to \(x\) is denoted as \(\frac{\partial w}{\partial x}\) and considers any change resulting from variations in \(x\) alone.


• Calculating a derivative means determining the rate of change or the slope of the tangent to the curve at any point.


By holding other variables constant, we specifically focus on how the function behaves relative to changes in that one variable.

Partial Differentiation

Partial differentiation is a technique used to find the derivative of a function of multiple variables with respect to one variable at a time. This analytical method provides insight into how each individual variable contributes to the change in the overall function.

• To perform a partial differentiation, you treat all other variables as constants while differentiating with respect to the variable of interest.


• In the given function \(w = x^2 - 3xy + 4yz + z^3\), the process involves differentiating separately with respect to \(x\), \(y\), and \(z\).


• The partial derivatives \(\frac{\partial w}{\partial x}\), \(\frac{\partial w}{\partial y}\), and \(\frac{\partial w}{\partial z}\) provide specific slopes in the direction of each variable.


This focused approach allows scientists and engineers to isolate and analyze the effect each variable has within more complex systems.