Problem 13 - Advanced Engineering Mathematics

Question

Find the first partial derivatives of the given function. $$ z=x^{2}-x y^{2}+4 y^{5} $$

Step-by-Step Solution

Answer

\( \frac{\partial z}{\partial x} = 2x - y^{2} \) and \( \frac{\partial z}{\partial y} = -2xy + 20y^{4} \).

Step 1: Identify the function and variables

The given function is \( z = x^{2} - x y^{2} + 4 y^{5} \). The variables are \( x \) and \( y \) since \( z \) depends on both.

Step 2: Differentiate with respect to x

To find the partial derivative of \( z \) with respect to \( x \) (denoted \( \frac{\partial z}{\partial x} \)), treat \( y \) as a constant and differentiate term by term.\[ \frac{\partial z}{\partial x} = \frac{\partial}{\partial x}(x^{2}) - \frac{\partial}{\partial x}(x y^{2}) + \frac{\partial}{\partial x}(4 y^{5}) \]- \( \frac{\partial}{\partial x}(x^{2}) = 2x \)- \( \frac{\partial}{\partial x}(x y^{2}) = y^{2} \)- \( \frac{\partial}{\partial x}(4 y^{5}) = 0 \) (since \( y \) is treated as a constant here)So, \( \frac{\partial z}{\partial x} = 2x - y^{2} \).

Step 3: Differentiate with respect to y

To find the partial derivative of \( z \) with respect to \( y \) (denoted \( \frac{\partial z}{\partial y} \)), treat \( x \) as a constant and differentiate term by term.\[ \frac{\partial z}{\partial y} = \frac{\partial}{\partial y}(x^{2}) - \frac{\partial}{\partial y}(x y^{2}) + \frac{\partial}{\partial y}(4 y^{5}) \]- \( \frac{\partial}{\partial y}(x^{2}) = 0 \) (since \( x \) is treated as a constant here)- \( \frac{\partial}{\partial y}(x y^{2}) = 2xy \)- \( \frac{\partial}{\partial y}(4 y^{5}) = 20y^{4} \)So, \( \frac{\partial z}{\partial y} = -2xy + 20y^{4} \).

Key Concepts

Multivariable Calculus

In multivariable calculus, we explore functions of multiple independent variables and their related derivatives. It extends the concepts of calculus from functions of a single variable to functions with more than one variable. Classically, you might have seen functions as a simple curve in a plane, while in multivariable calculus, they can be interpreted as surfaces in three-dimensional space.
In this context, our function \( z = x^2 - xy^2 + 4y^5 \) depends on two variables, \( x \) and \( y \). This means the variable \( z \) can present values based on different combinations of \( x \) and \( y \). This function represents a surface in a 3D space where each point along the surface is determined by the pair \( (x, y) \) and the resulting value of \( z \).
Understanding surfaces and their behavior through concepts such as slope in multivariable calculus can be challenging yet rewarding. Calculating partial derivatives is one technique to study changes in multivariable functions by focusing on one variable at a time, holding others constant. This helps to dissect how the surface curves and shifts along any axis of interest.

Differentiation

Differentiation is the process of finding the derivative, which is a measure of how a function changes as its input changes. With multivariable functions, we often find partial derivatives to understand the function's behavior with respect to each variable independently.
In our example, partial derivatives are calculated separately concerning \( x \) and \( y \). To find the partial derivative of \( z \) with respect to \( x \), denoted \( \frac{\partial z}{\partial x} \), we treat \( y \) as a constant and differentiate each term in the expression:

• The derivative of \( x^2 \) with respect to \( x \) is \( 2x \).
• For \( xy^2 \), since \( y^2 \) is constant, its derivative with respect to \( x \) is \( y^2 \).
• Finally, \( 4y^5 \) is independent of \( x \), so its derivative with respect to \( x \) is \( 0 \).

Thus, the partial derivative \( \frac{\partial z}{\partial x} = 2x - y^2 \).
This same process can be repeated for \( y \), treating \( x \) as a constant to find \( \frac{\partial z}{\partial y} = -2xy + 20y^4 \). The understanding of how each variable individually influences the function's output is critical in engineering and the sciences.

Engineering Mathematics

Engineering mathematics employs mathematical methods and techniques that are crucial in solving engineering problems. Multivariable calculus, including partial differentiation, is a key component in myriad fields such as fluid dynamics, thermodynamics, and electromagnetism.
Using partial derivatives, like in the exercise above, engineers can model and understand real-world systems where multiple variables interact. For example, if modelling the temperature distribution within a material, the same principles will apply where you might need to evaluate how temperature varies at different points, considering changes brought by heat sources or external conditions.
By comprehending the nature of partial derivatives — how one aspect of a system might change while another remains constant — engineers can make predictions, optimize systems for better efficiency, and solve complex differential equations that are common in engineering design and analysis.