Problem 273 - Calculus Volume 3

Question

For the following exercises, find the directional derivative of the function at point \(P\) in the direction of \(\mathbf{v}\). $$f(x, y)=y^{2}+x z, P(1,2,2), \quad \mathbf{v}=\langle 2,-1,2\rangle$$

Step-by-Step Solution

Answer

Directional derivative is \(2\).

Step 1: Apply differentiation rules

Use the appropriate differentiation rules (product, quotient, chain) to find the derivative.

Step 2: Simplify the result

The derivative is Directional derivative is \(2\)..

Key Concepts

Gradient

The gradient is a vector that points in the direction of the greatest rate of increase of a function. It is crucial in multivariable calculus, specifically when dealing with directional derivatives. The gradient of a function, denoted as \( abla f \), combines the partial derivatives of each variable into a single vector. This vector encapsulates how a function changes as each variable simultaneously changes.
For instance, if our function is \( f(x, y, z) = y^2 + xz \), the gradient \( abla f \) is calculated from the partial derivatives \( \frac{\partial f}{\partial x} = z \), \( \frac{\partial f}{\partial y} = 2y \), and \( \frac{\partial f}{\partial z} = x \). Thus, the gradient becomes a vector \((z, 2y, x)\).

• The direction of the gradient vector shows the quickest direction of increase of the function.
• The magnitude (or length) of the gradient vector indicates the rate of increase in that direction.
• To find the gradient at a point, substitute the point's coordinates into the gradient vector.

Understanding gradients is key to mastering directional derivatives, as they provide the necessary components to determine how functions behave in different directions.

Partial Derivatives

Partial derivatives are the cornerstone of understanding how multivariable functions change when each variable is varied individually. When dealing with functions of more than one variable, partial derivatives allow us to focus on one variable at a time, holding the other variables constant.
For the function \( f(x, y, z) = y^2 + xz \), the partial derivatives are computed as follows:

• \( \frac{\partial f}{\partial x} = z \) - This considers \( x \) as the variable while treating \( y \) and \( z \) as constants.
• \( \frac{\partial f}{\partial y} = 2y \) - Here, \( y \) is the variable, with \( x \) and \( z \) held constant.
• \( \frac{\partial f}{\partial z} = x \) - In this case, \( z \) is varied, keeping \( x \) and \( y \) constant.

Each partial derivative provides insight into how the function changes along each respective axis of its domain, and combined, they help form the overall gradient.

Multivariable Calculus

Multivariable calculus extends the concepts of calculus to functions of several variables. It plays a vital role in fields such as physics, engineering, and economics, where systems often depend on multiple changing factors.
Some vital concepts in multivariable calculus include:

• Gradients: These are crucial for optimization and understanding how functions change in multidimensional spaces.
• Directional Derivatives: These help in finding the rate of change of a function in any given direction, not just along an axis.
• Partial Derivatives: These allow for the breakdown and analysis of multi-variable functions step-by-step by focusing on one variable at a time.

By using tools like gradients and partial derivatives, multivariable calculus equips us to tackle complex problems involving functions of more than one variable. The directional derivative, which combines these concepts, assesses changes in any specific direction, leading to powerful applications in almost any real-world system.