Problem 39 - Calculus Early Transcendentals: Pearson New International Edition

Question

If \(f(x, y, z)=3 x^{2} y-x y z+y^{2} z^{2}\), find each of the following: (a) \(f_{x}(x, y, z)\) (b) \(f_{y}(0,1,2)\) (c) \(f_{x y}(x, y, z)\)

Step-by-Step Solution

Answer

(a) \(f_x(x, y, z) = 6xy - yz\); (b) \(f_y(0,1,2) = 8\); (c) \(f_{xy}(x, y, z) = 6x - z\).

Step 1: Find Partial Derivative with Respect to x

To find the partial derivative \(f_x(x, y, z)\), differentiate the function \(f(x, y, z) = 3x^2y - xyz + y^2z^2\) with respect to \(x\). Treat \(y\) and \(z\) as constants. Using the power rule, the derivative of \(3x^2y\) with respect to \(x\) is \(6xy\). The derivative of \(-xyz\) with respect to \(x\) is \(-yz\), and the derivative of \(y^2z^2\) with respect to \(x\) is 0. Therefore, \(f_x(x, y, z) = 6xy - yz\).

Step 2: Evaluate Partial Derivative at a Point

To evaluate \(f_y(0,1,2)\), first find \(f_y(x, y, z)\). Differentiate \(f(x, y, z)\) with respect to \(y\), treating \(x\) and \(z\) as constants. The derivative of \(3x^2y\) is \(3x^2\), of \(-xyz\) is \(-xz\), and of \(y^2z^2\) is \(2yz^2\). Thus, \(f_y(x, y, z) = 3x^2 - xz + 2yz^2\). Now substitute \((x, y, z) = (0, 1, 2)\) into \(f_y(x, y, z)\): \(f_y(0, 1, 2) = 3(0)^2 - 0 \cdot 2 + 2 \cdot 1 \cdot (2)^2 = 0 - 0 + 8 = 8\).

Step 3: Find Second Mixed Partial Derivative

For \(f_{xy}(x, y, z)\), differentiate the partial derivative \(f_x(x, y, z) = 6xy - yz\) with respect to \(y\). When differentiating \(6xy\) with respect to \(y\), we get \(6x\), and differentiating \(-yz\) with respect to \(y\) gives \(-z\). Hence, \(f_{xy}(x, y, z) = 6x - z\).

Key Concepts

Partial Derivatives

Partial derivatives are key in multivariable calculus. They represent how a function changes as each variable changes while holding the other variables constant.
When we calculate a partial derivative, we essentially "zoom in" on one variable and see how the function behaves in that particular direction.

• For our function, we took the derivative of each term involving the variable of interest, while treating other variables as constants.
• For example, in the function \(f(x, y, z) = 3x^2y - xyz + y^2z^2\), to find \(f_x\), we differentiated \(3x^2y\), \(-xyz\), and \(y^2z^2\) with respect to \(x\), resulting in \(6xy - yz\).

Partial derivatives are useful because they allow us to explore rates of change in one dimension in the context of a multi-variable system. This concept is widely applied in fields such as physics for understanding gradients and in economics for analyzing changes in demand.

Derivative Rules

When finding derivatives, we use several rules that simplify the process, especially with more complex expressions involving multiple variables. Here are some important rules you should know:

• Power Rule: This rule states that the derivative of \(x^n\) is \(nx^{n-1}\). It's useful for differentiating polynomials.
• Constant Rule: The derivative of a constant is zero. This helps when handling terms where variables are fixed.
• Sum Rule: The derivative of a sum is the sum of the derivatives. This allows us to break down complicated expressions into simpler parts.

In our specific problem, using these rules makes it easier to find partial derivatives. For instance, while differentiating with respect to \(x\), the term \(y^2z^2\) is treated as a constant, hence its derivative is zero. Recognizing where to apply these rules can significantly reduce the complexity involved in taking derivatives of multi-variable functions.

Mixed Partial Derivatives

Mixed partial derivatives involve taking the derivative of a function twice, each time with respect to a different variable. They provide insight into how variables together influence a function.
To compute a mixed partial derivative:

• First, take the partial derivative of the function with respect to one variable.
• Then, take the derivative of the resulting expression with respect to another variable.

In our context, \(f_{xy}(x, y, z)\) means we first differentiate \(f(x, y, z)\) with respect to \(x\) to obtain \(6xy - yz\), then differentiate this result with respect to \(y\), giving \(6x - z\).
Mixed partial derivatives can reveal complexities in functions, like interactions between variables, and are used in optimization problems and in understanding the behavior of surfaces described by functions.