Problem 7 - Calculus with Concepts in Calculus

Question

Find the gradient of the function. $$ g(x, y, z)=\frac{-x+y}{-x+z} $$

Step-by-Step Solution

Answer

The gradient is \( \nabla g(x, y, z) = \left( \frac{x-y+z}{(-x+z)^2}, \frac{1}{-x+z}, \frac{x-y}{(-x+z)^2} \right) \).

Step 1: Identify the Function Components

Given the function is \( g(x, y, z) = \frac{-x + y}{-x + z} \). It is a function of three variables: \(x\), \(y\), and \(z\). To find the gradient, we need the partial derivatives with respect to each of these variables.

Step 2: Partial Derivative with Respect to x

To find the partial derivative of \( g \) with respect to \( x \), apply the quotient rule: \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \).Here, \( u = -x + y \) and \( v = -x + z \). Hence, \( \frac{du}{dx} = -1 \) and \( \frac{dv}{dx} = -1 \).\[ \frac{\partial g}{\partial x} = \frac{(-x+z)(-1) - (-x+y)(-1)}{(-x+z)^2} = \frac{x-y+z}{(-x+z)^2} \]

Step 3: Partial Derivative with Respect to y

Finding the partial derivative of \( g \) with respect to \( y \) is straightforward as \( y \) only appears in the numerator.\[ \frac{\partial g}{\partial y} = \frac{\partial}{\partial y} \left( \frac{-x+y}{-x+z} \right) = \frac{1}{-x+z} \]

Step 4: Partial Derivative with Respect to z

Now, find the partial derivative of \( g \) with respect to \( z \). This uses the basic derivative rule since \( z \) only appears in the denominator.\[ \frac{\partial g}{\partial z} = \frac{\partial}{\partial z} \left( \frac{-x+y}{-x+z} \right) = \frac{-(-x+y)}{(-x+z)^2} = \frac{x-y}{(-x+z)^2} \]

Step 5: Combine the Partial Derivatives into the Gradient Vector

The gradient of \( g \) is the vector of its partial derivatives with respect to each variable. Therefore, the gradient is given by:\[ abla g(x, y, z) = \left( \frac{x-y+z}{(-x+z)^2}, \frac{1}{-x+z}, \frac{x-y}{(-x+z)^2} \right) \] This represents the direction and rate of the steepest ascent of the function.

Key Concepts

Partial Derivatives

When dealing with multivariable functions like \[ g(x, y, z) = \frac{-x + y}{-x + z} \], it is crucial to understand partial derivatives. These are derivatives that consider how a function changes as only one variable changes, keeping others constant. Imagine having a piece of fabric, and you pull at just one corner; the change or tension you see represents a partial derivative in the context of that direction.
To find the gradient of the function, we compute the partial derivatives with respect to each variable: \( x \), \( y \), and \( z \).

• Partial with respect to \( x \): Uses the quotient rule (discussed later), given both \( x \) is in the numerator and denominator.
• Partial with respect to \( y \): Is simpler as \( y \) only appears in the numerator, leading to a straightforward derivative.
• Partial with respect to \( z \): Focuses on \( z \) in the denominator, altering the derivative approach.

These partial derivatives represent the slope of the function in a specific direction along each variable's axis, giving insights into how the function behaves locally.

Vector Calculus

Vector calculus is a field of mathematics that extends calculus concepts to vector fields. In the exercise, we dealt with a scalar function in three-dimensional space. The gradient, denoted by \( abla g \), turns this scalar into a vector.
The gradient vector provides vital information about the function's behavior:

• The direction of the gradient indicates the direction of the steepest ascent.
• The magnitude of the gradient reveals how fast the function increases.

For the function \( g(x, y, z) \), the gradient is a vector composed of its partial derivatives:\[abla g(x, y, z) = \left( \frac{x-y+z}{(-x+z)^2}, \frac{1}{-x+z}, \frac{x-y}{(-x+z)^2} \right)\]This combines information from each variable, showing how changes in \( x \), \( y \), and \( z \) affect the function together.
Through vector calculus, we can tackle complex systems and understand interactions within multivariable contexts.

Quotient Rule for Derivatives

The quotient rule is essential when differentiating a function that is the ratio of two functions. It provides a method to find the derivative of a division such as \[ g(x, y, z) = \frac{-x + y}{-x + z} \].
The formula for the quotient rule in the context of a function \( \frac{u}{v} \) is:\[\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}\]Here, \( u = -x + y \) and \( v = -x + z \).

• Derivative of \( u \) with respect to \( x \): Simply \( -1 \), since it’s linear.
• Derivative of \( v \) with respect to \( x \): Also \( -1 \).

Substituting these back into the rule gives the partial derivative with respect to \( x \) as:\[\frac{\partial g}{\partial x} = \frac{(-x+z)(-1) - (-x+y)(-1)}{(-x+z)^2} = \frac{x-y+z}{(-x+z)^2}\]This systematic approach can handle any function division, simplifying even complex expressions.