Question

In each case, given \(z=f(x, y)\) find \(z_{x}\) and \(z_{y}\). (a) \(z=x y\) (b) \(z=3 x y\) (c) \(z=-9 y x\) (d) \(z=x^{2} y\) (e) \(z=9 x^{2} y\) (f) \(z=8 x y^{2}\)

Short answer

Question: Find the partial derivatives of the following functions: a) \(z = xy\) b) \(z = 3xy\) c) \(z = -9yx\) d) \(z = x^2y\) e) \(z = 9x^2y\) f) \(z = 8xy^2\) Answer: a) \(z_x = y\), \(z_y = x\) b) \(z_x = 3y\), \(z_y = 3x\) c) \(z_x = -9y\), \(z_y = -9x\) d) \(z_x = 2xy\), \(z_y = x^2\) e) \(z_x = 18xy\), \(z_y = 9x^2\) f) \(z_x = 8y^2\), \(z_y = 16xy\)

Step-by-step solution

Identify the function

Here, the function is given as \(z = xy\).

Find the partial derivatives

To find the partial derivatives \(z_x\) and \(z_y\), take the derivative of the function with respect to x and y, keeping the other variable constant. \(z_x = \frac{\partial z}{\partial x} = \frac{\partial (xy)}{\partial x} = y\) \(z_y = \frac{\partial z}{\partial y} = \frac{\partial (xy)}{\partial y} = x\) #b)

Identify the function

Here, the function is given as \(z = 3xy\).

Find the partial derivatives

To find the partial derivatives \(z_x\) and \(z_y\), take the derivative of the function with respect to x and y, keeping the other variable constant. \(z_x = \frac{\partial z}{\partial x} = \frac{\partial (3xy)}{\partial x} = 3y\) \(z_y = \frac{\partial z}{\partial y} = \frac{\partial (3xy)}{\partial y} = 3x\) #c)

Identify the function

Here, the function is given as \(z = -9yx\).

Find the partial derivatives

To find the partial derivatives \(z_x\) and \(z_y\), take the derivative of the function with respect to x and y, keeping the other variable constant. \(z_x = \frac{\partial z}{\partial x} = \frac{\partial (-9yx)}{\partial x} = -9y\) \(z_y = \frac{\partial z}{\partial y} = \frac{\partial (-9yx)}{\partial y} = -9x\) #d)

Identify the function

Here, the function is given as \(z = x^2y\).

Find the partial derivatives

To find the partial derivatives \(z_x\) and \(z_y\), take the derivative of the function with respect to x and y, keeping the other variable constant. \(z_x = \frac{\partial z}{\partial x} = \frac{\partial (x^2y)}{\partial x} = 2xy\) \(z_y = \frac{\partial z}{\partial y} = \frac{\partial (x^2y)}{\partial y} = x^2\) #e)

Identify the function

Here, the function is given as \(z = 9x^2y\).

Find the partial derivatives

To find the partial derivatives \(z_x\) and \(z_y\), take the derivative of the function with respect to x and y, keeping the other variable constant. \(z_x = \frac{\partial z}{\partial x} = \frac{\partial (9x^2y)}{\partial x} = 18xy\) \(z_y = \frac{\partial z}{\partial y} = \frac{\partial (9x^2y)}{\partial y} = 9x^2\) #f)

Identify the function

Here, the function is given as \(z = 8xy^2\).

Find the partial derivatives

To find the partial derivatives \(z_x\) and \(z_y\), take the derivative of the function with respect to x and y, keeping the other variable constant. \(z_x = \frac{\partial z}{\partial x} = \frac{\partial (8xy^2)}{\partial x} = 8y^2\) \(z_y = \frac{\partial z}{\partial y} = \frac{\partial (8xy^2)}{\partial y} = 16xy\)

Key concepts

Multivariable Calculus

Multivariable calculus is an extension of single-variable calculus. It deals with functions that have more than one variable. This allows us to explore how changes in multiple inputs affect the output. Imagine a surface in 3-dimensional space. Here, the height of the surface can depend on the position in the two-dimensional plane beneath it. We use multivariable functions to describe such surfaces mathematically.In multivariable calculus, we study limits, integration, and differentiation with respect to functions of several variables. Most engineering and physical problems rely on these concepts to understand phenomena that depend on multiple factors. The function given in our problem is a simple example. It is a function of two variables \(x\) and \(y\), and the output \(z\) is dependent on both.

Partial Differentiation

Partial differentiation focuses on finding the derivative of a function with respect to one variable while keeping others constant. This is essential in multivariable calculus because it gives us insight into how each variable separately influences the output. For a function \(z = f(x, y)\), the partial derivative with respect to \(x\) is denoted as \(z_x\). This derivative shows the rate of change of \(z\) as \(x\) changes, while \(y\) remains constant. Similarly, the partial derivative with respect to \(y\), \(z_y\), describes the change in \(z\) as \(y\) changes, keeping \(x\) constant.To compute partial derivatives, treat the variable of differentiation normally, while treating the other variables as constants.

• For \(z=x y\), \(z_x=y\) and \(z_y=x\).
• Different constants, both multiplied by variables, act similarly. For \(z=3xy\), \(z_x = 3y\) and \(z_y = 3x\).

Understanding partial derivatives helps unravel how systems respond to changes in particular dimensions, which is crucial in engineering and physics.

Engineering Mathematics

In engineering, mathematics serves as a crucial tool and language. It allows complex systems to be modeled and understood. Partial derivatives are particularly useful in fields like fluid dynamics, thermodynamics, and structural analysis. When engineers design systems, they often need to assess how each parameter affects a whole system. Partial differentiation enables them to isolate impacts and optimize designs efficiently. For example, in heat transfer analysis, a temperature distribution function depends on various coordinates. By taking partial derivatives, we effectively measure the sensitivity of temperature to positions or time. Moreover, these concepts extend to electrical circuits, mechanical engineering, and beyond. Engineers use calculus to predict system behavior under varying conditions, ensuring the reliability and efficiency of their designs. By mastering these mathematical tools, engineers can innovate and solve real-world challenges effectively.